Predicting Choice from Information Costs *Elliot Lipnowski†

Doron Ravid‡

Columbia University

University of Chicago

August 11, 2023

Abstract
An agent acquires a costly flexible signal before making a decision. We explore to what
degree knowledge of the agent’s information costs helps predict her behavior. We establish an
impossibility result: learning costs alone generate no testable restrictions on choice without
also imposing constraints on actions’ state-dependent utilities. By contrast, choices from amenu often uniquely pin down the agent’s decisions in all submenus. To prove the latter
result, we define iteratively differentiable cost functions, a tractable class amenable to firstorder techniques. Finally, we construct tight tests for a multi-menu data set to be consistent
with a given cost.

1. Introduction
Sparked by Sims’ (1998; 2003; 2006) studies on rational inattention, the last two decades have seena growing interest in models of costly flexible learning. Compared with the traditional frameworkof decision-making under uncertainty, these models postulate that the agent’s behavior depends
on an additional parameter: the cost of information acquisition. The appropriate values for thisparameter have been the subject of intense inquiry. Some studies, such as Caplin and Dean (2015),
de Oliveira et al. (2017), Dean and Neligh (2022), Dewan and Neligh (2020), and Caplin et al.
(2020), seek to answer this question empirically, developing tools for identifying, measuring, andtesting the agent’s cost function in experiments. Other studies take an axiomatic approach, advocating for classes of parameterized cost functions based on their characterizing features (e.g., Pomatto,* We would like to thank Ben Brooks, Mark Dean, Laura Doval, Tommaso Denti, Mira Frick, Daniel Gottlieb,

Ryota Iijima, Navin Kartik, R. Vijay Krishna, Ce Liu, Daniel Martin, Stephen Morris, Pietro Ortoleva, John Rehbeck,
Phil Reny, Marzena Rostek, Kareen Rozen, Pablo Schenone, and Omer Tamuz for their comments. Tianhao Liu
provided excellent research assistance.
†
e.lipnowski@columbia.edu‡
dravid@uchicago.edu

1Strack, and Tamuz, 2020; Hébert and Woodford, 2021a; Caplin, Dean, and Leahy, 2021). An alternative string of papers explores which cost functions can be microfounded via dynamic learning(e.g., Morris and Strack, 2019; Bloedel and Zhong, 2020; Hébert and Woodford, 2021b) or as aphysical cost of performing experiments (Denti et al., 2021). In this paper, we ask a complementaryquestion: In what ways does the agent’s information-acquisition cost restrict her behavior?
There are several reasons why one might be interested in our question. First, as explained
above, multiple existing studies are dedicated to uncovering facts about information acquisition
costs. By studying the restrictions these costs put on behavior, we outline the boundaries of said research program. Second, our question is relevant for applied modelers interested in understanding
the connection between cost-based assumptions and behavior. For example, Proposition 2 shows
our analysis yields an immediate characterization of the set of cost functions satisfying “continuous choice”, a property which has economic implications (Morris and Yang, 2021). And third, ourquestion can pave the path to a new way of testing whether a behavioral data set is consistent witha given cost function, as we demonstrate in section 6.
We study a canonical flexible-learning model in which an agent chooses from a finite set ofalternatives, the benefits from which depend on a stochastic state. Before making her decision,
the agent chooses what signal to acquire about this state. Learning comes at a cost, which theagent subtracts from the expected benefit she derives from her final decision. After choosing her
information, the agent observes a signal realization and takes an action. Following the literature(Caplin and Dean, 2015), we refer to the resulting probability in which the agent takes each action
conditional on the state as the agent’s stochastic choice rule (SCR).
We begin by studying one’s ability to forecast the agent’s choices using only her cost of information acquisition. Theorem 1 shows this exercise is essentially futile: holding the agent’s
costs fixed, one can approximate any finite-cost SCR arbitrarily well with SCRs that are uniquely
optimal for some utility function. Therefore, other than ruling out some SCRs as technologicallyinfeasible, the agent’s learning costs make no predictions that could be falsified with finite data.
The above conclusion has implications regarding the connection between cost-based assumptions and behavior. For example, Morris and Yang (2021) introduce the continuous choice property,
which says the agent’s optimal SCR must vary continuously with the state regardless of her objective function. Morris and Yang (2021) show this property leads to significant restrictions on thekind of equilibria that may arise in regime-change games when learning costs are small. Our results imply the continuous choice property can hold only for cost functions that make discontinuous
SCRs infeasible; that is, the cost function must assign such SCRs an infinite cost (Proposition 2).
More generally, Theorem 1 implies the agent’s cost function on its own imposes few restrictions
on the agent’s behavior in a single menu. However, we show the agent’s costs can significantly
restrict the way the agent behaves across menus. To make this point, we focus on a novel class of2

smooth cost functions that we call iteratively differentiable. We prove such cost functions allow
one to solve for the agent’s optimal SCR given a fixed utility function using a first-order condition(Lemma 3). One can also apply this condition to do the reverse, namely, to solve for the utility
functions that make a fixed SCR optimal. In Theorem 2, we use the ability to invert utility fromchoices to show that, for finite-valued iteratively differentiable costs satisfying an infinite-slope
condition, one can often use the agent’s actions in one menu to pin down her behavior in all submenus. Thus, under the theorem’s conditions, once the analyst knows the agent’s cost of learning,
the agent’s choice when facing the grand menu is the sole remaining degree of freedom in theagent’s behavior.
That the agent’s cost function restricts her behavior across menus is useful not only for predicting her actions, but also for testing whether or not her decisions are consistent with a given
cost function. In section 6 we develop such a test, assuming the given cost function is iteratively
differentiable and finite-valued. Our test takes as its starting point a data set describing the agent’s
chosen SCR in a collection of menus. From this data set, our test constructs a bipartite graph withnodes labeled by actions and menus, and with an edge between an action and a menu if the agent
sometimes takes said action when faced with that menu. We identify each cycle in this graph witha set of equations, and show these equations are satisfied for all cycles if and only if the data set iscompatible with the agent having a common utility and the pre-specified cost function. Moreover,it suffices to check the equations associated with a small set of cycles, called a cycle basis. We also
prove our test is tight for cost functions satisfying the prerequisites of Theorem 2: testing the equations corresponding to any collection of cycles that does not include a cycle basis is insufficient forproving consistency of a data set.
Several studies consider the problem of testing whether a data set is consistent with costly information acquisition, and if so, whether one can use these data to make inferences about the agent’s
cost of information (e.g., Caplin and Dean, 2015; Caplin, Dean, and Leahy, 2017; de Oliveira
et al., 2017; Caplin et al., 2020; Chambers, Liu, and Rehbeck, 2020; Dewan and Neligh, 2020;
Denti, 2022; Lin, 2022). This literature typically assumes that, in addition to observing the agent’s
choices, the analyst also knows the agent’s utility function, which is either observed directly, orinferred from decision problems in which information acquisition plays no role. Our paper contributes to this literature by delineating the kind of data needed for testing various hypotheses
regarding the agent’s information-acquisition costs. In particular, we show that without utility information, one cannot test such hypothesis using the agent’s behavior in a single menu. We also
provide a method for using variation in the agent’s menu to check whether or not the agent faces agiven hypothetical cost function.
Other papers have advocated the use of various cost functions based on non-behavioral justifications. For example, Mensch (2018), Pomatto, Strack, and Tamuz (2020), and Hébert and3

Woodford (2021a) pin down particular functional forms for the agent’s learning costs by stating
properties directly on the agent’s cost function rather than on the decisions that it generates.1 Other
papers, such as Morris and Strack (2019), Bloedel and Zhong (2020), and Hébert and Woodford(2021b), pin down specific functional forms by asking which static cost functions can be microfounded as coming from a dynamic learning process.2 Our paper complements this literature by
studying the restrictions imposed by a given cost function, and providing a blueprint for generating
predictions and testing hypotheses on the agent’s information acquisition costs without the needfor a comprehensive revealed preference analysis.
We view our main substantive results to be Theorem 1, Proposition 2, Theorem 2, and Corollary 1. En route, we prove some technical results on models with costly information acquisition,
including: an algebraic sufficient condition for an optimal SCR to be uniquely optimal (Proposition 9); a Langrangian first-order condition for optimality of a given SCR under smooth costs(Proposition 3); a reduction of the agent’s problem with differentiable costs to her problem undera posterior-separable approximation (Lemma 3); and a generic uniqueness result for optimal SCRs
(Lemma 4). The key to these results is the observation that the behavioral content of the agent’s
information costs is encoded in the subdifferential of the indirect cost function; that is, the functionthat gives the minimal cost of implementing each SCR. Because this function is well-defined (and,we show, well-behaved) rather generally, our results apply to cost functions and environments notcovered by the extant literature. Thus, in addition to studying the behavioral content of information
costs, our paper also provides technical tools that may be useful in future work.

2. ModelAn agent makes a decision from a finite set A of actions with |A| > 1. The payoff from each action
depends on a payoff state ω belonging to some compact metric space Ω and distributed according
to some probability measure µ0 ∈ ∆Ω.3 Without loss of generality, we take µ0 to have full support.
The agent’s utility from choosing action a ∈ A in state ω ∈ Ω is ua (ω). We assume ua ∈ L1 (µ0 ),meaning the agent’s expected payoff E [ua ] from every action a ∈ A is well defined, and refer toAu := (ua )a∈A ∈ U := L1 (µ0 ) as the agent’s utility function.4
1

Relatedly, Frankel and Kamenica (2019) study axiomatically the appropriate measures of the ex-post value of
information.
2
A related study is Denti et al. (2021), which obtain conditions under which a cost function defined over the agent’s
distribution of posterior beliefs for all priors can be micro-founded as a prior-independent cost over experiments.
3
View any Polish space Y as a measurable space, endowed with its Borel sigma-algebra. Let ∆Y denote the set ofprobability measures on this measurable space, interpreting linear combinations in ∆Y pointwise. Unless otherwise
stated, we endow ∆Y with the weak* topology generated by continuous bounded functions, which in turn makes ∆Y
a compact metrizable space if Y is compact. We let supp(γ)
denote the support of γ for any γ ∈ ∆Y .
R
4For any random variable f : Ω → R, let E[f ] := f (ω) µ0 (dω) whenever the latter Lebesgue integral is welldefined. Recall the space L1 (µ0 ) is the set of all measurable functions f : Ω → R such that E|f | < ∞, modulo

4Information Policies. Before taking her action, our agent chooses what signal to observe in orderto learn about ω. We assume our agent is Bayesian, and model the agent’s information via the
distribution of her posterior beliefs, p ∈ ∆∆Ω. Specifically, we allow the agent to choose any
R
p whose average equals the prior, µ p(dµ) = µ0 , which is equivalent to letting her select any
p that originates from some signal structure.5 We refer to any p that averages back to the prioras an information policy, and denote the set of all information policies by P. Occasionally, we
will refer to the set P F ⊆ P of simple information policies, namely those with finite support.
Relatedly, a simply drawn posterior is a posterior that belongs to the support of some simple
information policy.
We order information policies via informativeness in the sense of Blackwell (1953). Specifically, we say p ∈ P is more informative than q ∈ P (written as p  q) if p is a mean-preserving
spread of q.6 An information policy p is strictly more informative than q (written p  q) if p  q
and p , q. Intuitively, p is more informative than q if observing p is equivalent to observing q andan additional signal. For a review of the connection between this information ranking and othernotions of informativeness, see Khan et al. (2019).
Information Costs. Information comes at a cost that is summarized via a function taking values
in the extended nonnegative reals,
C : P → R+ := R+ ∪ {∞}.
By letting C take a value of infinity, we allow it to encode constraints on the set of feasible information policies. We assume C is a lower-semicontinuous convex function that is proper—that is, notglobally equal to ∞—and let P C = C −1 (R) denote its (effective) domain, namely, the nonempty
set of information policies that can be induced at finite cost. We also require C to be monotone,
meaning C(p) ≥ C(q) whenever p  q. Sometimes, we focus on costs that are strictly monotone,by which we mean C(p) > C(q) holds whenever C(q) is finite and p  q. As we explain in
section 7, assuming C is convex and monotone is without loss of generality.
The Agent’s Problem. After choosing her information, the agent observes her signal realizationand takes an action. An action strategy is a measurable mapping, α : ∆Ω → ∆A, where α(a|µ)
is the probability the agent chooses action a when her posterior belief is µ. A strategy is an
information policy p paired with an action strategy α. The agent’s payoff from the strategy (p, α),µ0 -almost sure equivalence, equipped with the L1 (µ0 ) norm, kf k1 := E|f |.
5See, for example, Aumann and Maschler (1995), Kamenica and Gentzkow (2011), or Benoı̂t and Dubra (2011).
For the sake of completeness, Appendix C.2 provides a proof that applies
to the case of infinite states.
R6µ̃m(dµ̃|µ)= µ for all µ ∈ ∆Ω, and p(D) =
That
is,
some
measurable
m
:
∆Ω
→
∆∆Ω
existssuchthatRm(D|µ) q(dµ) for all Borel D ⊆ ∆Ω.

5

given benefit u ∈ U, is
Z

Z

∆Ω

A

ua α(da|µ) p(dµ) − C(p).

Observe some strategy yields a finite value, because some information policy yields finite cost. We
say (p, α) is u-optimal if it maximizes the above objective among all possible strategies.
Stochastic Choice Rules. We follow Caplin and Dean (2015) and summarize the agent’s behavior
via a state-dependent stochastic choice rule. Within our more general setting, such a rule consistof a vector of (essentially bounded) mappings,
s = (sa )a∈A ,where sa ∈ L∞ (µ0 ) gives the conditional probability the agent takes action a given the state.7
P
Because probabilities are positive and add up to 1, we must have sa ≥ 0 for all a and a∈A sa =1. We refer to every s ∈ S := L∞ (µ0 )A that satisfies these constraints as a (state-dependent)
stochastic choice rule (SCR),8 and denote the set of all SCRs by S.
The support of an SCR s is the set of actions it generates with positive probability, supp s ={a ∈ A : sa , 0}. An SCR s has full support if it uses all actions, that is, if supp s = A. The SCR
has conditionally full support if it uses all actions in all states, meaning sa is µ0 -almost surely
strictly positive for every a ∈ A.
A strategy (p, α) induces an SCR s if for every action a ∈ A and event Ω̂ ⊆ Ω,Zα(a|µ) µ(Ω̂) p(dµ) = E [1Ω̂ sa ] .

An information policy p ∈ P can induce s ∈ S if p can describe the information the agent receives
in some strategy that results in s, that is, if (p, α) induces s for some α.
Rationalizability We are interested in understanding which stochastic choice rules are optimal fora given objective, which are optimal for some objective, and which can be uniquely optimal. Givena utility function u ∈ U, we say s is u-rationalizable if it is induced by some u-optimal strategy(p, α). A stochastic choice rule is uniquely u-rationalizable if it is the only u-rationalizable
SCR. An SCR that is u-rationalizable for some u is rationalizable. We also say s is uniquely
Recall L∞ (µ0 ) is the space of all measurable functions f : Ω → R (identified up to µ0 -almost sure equality) thatare bounded µ0 -almost surely, equipped with the L∞ (µ0 ) norm, kf k∞ := ess sup |f |.
8
Because A is finite, the probability sa (ω) of action a conditional on state ω is well-defined up to µ0 -almost sureAequivalence by Radon-Nikodym. Consequently, a stochastic choice rule naturally lives in S = L∞ (µ0 ) , which
does not distinguish between functions that agree µ0 -almost surely. An equivalent formalism (given a disintegration
theorem) has stochastic choice rules living in the set of probability measures over A × Ω with marginal µ0 on Ω.
7

6

rationalizable if it is uniquely u-rationalizable for some u.9

Example Cost Functions
In this subsection, we provide some example cost functions satisfying our hypotheses.
Example 1 (Mutual Information Costs). Let K : ∆Ω → R be the Kullback-Leibler divergencefrom the prior µ0 ,K(µ) =

R log dµ (ω) µ(dω)

if µ  µ0 ,∞otherwise.

dµ0

By Posner (1975), K is lower semicontinuous. The mutual information cost function is given by
C(p) =

Z

K(µ) p(dµ).

This cost function was first introduced by Sims (1998; 2003; 2006) and has served as the workhorsecost function of the literature on rational inattention. See Csiszár (1974), Matějka and McKay(2015), Caplin, Dean, and Leahy (2019), and Denti, Marinacci, and Montrucchio (2020) for characterizations of optimal behavior under mutual information costs.
Example 2 (Log-Likelihood Ratio Costs). Pomatto, Strack, and Tamuz (2020) proposed and axiomatized the Log-likelihood ratio (LLR) cost function. The LLR cost function is defined for a finite
such that θ ω,ω = 0 for allΩ, and is parameterized by a matrix of positive numbers, θ ∈ RΩ×Ω+
LLR
ω ∈ Ω. Given θ, define the function cθ : ∆Ω → R via
cLLR
(µ) =
θ

X

θω,ω0

ω,ω 0

µ(ω)
µ(ω)
log
.
µ0 (ω)µ(ω 0 )

Then the θ-LLR cost function is given by
CθLLR (p) =

Z h

i

cLLR
(µ) − cLLR(µ0 ) p(dµ).
θθ

Example 3 (Posterior Separable Costs). Posterior separable costs are a generalization of mutual
information costs introduced by Caplin, Dean, and Leahy (2017). Let C be the set of convex, lower
semicontinuous functions from ∆Ω to R that assign µ0 a finite nonnegative value.10 We say C is9

In other words, s is uniquely rationalizable if some u exists such that s is the only u-rationalizable SCR; thisnotion is distinct from requiring that u be the unique utility that rationalizes s.
10Given the other assumptions on elements of C, that c(µ0 ) is finite is equivalent to the induced information cost C
being proper. Also note, when Ω is finite and c is finite, c is lower semicontinuous if and only if it is continuous.

7

posterior separable if some c ∈ C exists such that every p ∈ P has
C(p) =

Zc(µ) p(dµ).

In addition to mutual information (Example 1), the class of posterior separable costs includes
the log-likelihood ratio cost function from Example 2, and the neighborhood-based cost function
studied by Hébert and Woodford (2021a).
Example 4 (Transformed Costs). Let ψ : R+ → R+ be a nondecreasing, proper, convex, lower
semicontinuous function. Then,ZC(p) = ψc(µ) p(dµ)

satisfies our assumptions for any c ∈ C.
Example 5 (Quadratic Costs). Let c̃ : ∆Ω × ∆Ω → R be a symmetric, lower semicontinuousfunction that is convex in each argument and has c̃(µ0 , µ0 ) ≥ 0. Let
C(p) =Z Z

c̃(µ̃, µ) p(dµ̃) p(dµ).

Then, C satisfies our assumptions as long as c̃ is positive semidefinite, namely, as long as
Z Zc̃(µ̃, µ) (p − q)(dµ̃) (p − q)(dµ) ≥ 0

holds for all p, q ∈ P.
Example 6 (Maximum over a Set). Let C˜ ⊆ C be a compact (with respect to the supremum norm)
set of continuous functions. Then,
C(p) = max

Z

c∈C˜

c(µ) p(dµ)

satisfies our assumptions.

3. Cost Minimization
We begin our analysis by studying the cheapest way to induce a given SCR. Specifically, we solve
κ(s) = inf C(p) s.t. p can induce s.
p∈P

8Note s can be rationalizable only if the above program admits a solution: otherwise, no (p, α) thatinduces s can ever be optimal, because one can always attain the same utility with lower costs.
To solve the cost-minimization program, observe that every s can be viewed as a signal structure whose realizations take the form of recommended actions. Therefore, one can apply Bayes
rule to transform any s into its associated information policy, ps ∈ P. Formally, let
psa := E [sa ]be the ex-ante probability that s generates the recommendation a. Whenever psa > 0, Bayes’ rule
dictates the agent’s posterior belief µsa conditional on the realized action recommendation being a
is given by11sa (ω)µ0 (dω).
µsa (dω) =psa
It follows one can write ps as
ps =X

psa δµsa ,a∈Awhere δµsa ∈ ∆∆Ω denotes the distribution that generates µsa with probability 1. If every a isassociated with a different µsa , then ps is the information policy that yields the posterior µsa withprobability psa . We follow the literature (Caplin and Dean, 2015) and refer to µsa as the revealed
posterior of a given s, and ps as s’s revealed information policy.
Lemma 1. Policy p ∈ P can induce s ∈ S if and only if p  ps . Therefore, κ(s) = C(ps ).
Similar results are prevalent in the literature for less general settings (e.g., Caplin and Dean,
2015). Despite the difference in generality, the intuition is identical. If (p, α) induces s, one cangenerate ps by first drawing a posterior-action pair (µ, a) according to (p, α) and then revealing
only the realized action to the agent. Clearly, seeing a alone is less informative than seeing both aand µ. But because a is independent of the state conditional on µ, seeing µ and a together is justas informative as observing µ on its own. In other words, p is more informative than ps . Thus, thelowest-cost way of inducing s is given by ps .
Our next lemma uses the connection between SCRs and their revealed information policies toestablish several useful properties of the agent’s indirect cost function κ.
Lemma 2. The indirect cost κ is proper, convex, and weak* lower semicontinuous.
That κ is proper follows directly from existence of a finite-cost information policy. To provelower semicontinuity, we establish that weak* convergence of SCRs implies convergence of their
11

When psa = 0, we can let µsa ∈ ∆Ω be arbitrary wherever the term appears.

9revealed information policies. To show κ is convex, we establish that the information policy revealed by a convex combination of two SCRs is less informative than the convex combination of
the two policies. Using this fact, one can show κ is convex using convexity and monotonicity ofC.
We now rephrase the agent’s problem so that it takes the agent’s SCR s as its decision variable.
Given u, the benefit from choosing s is given by12"

E[u · s] = E#Xu a sa .

a∈A

Because κ(s) gives the minimum cost at which the agent obtains s, we get that s is u-rationalizable
if and only if it solves the programhimax E[u · s] − κ(s) .
s∈S

(1)

Observe the above program always admits a solution: because κ is weak* lower semicontinuous,
the program maximizes a weak*-upper-semicontinuous objective over a weak*-compact set.13 In
other words, every utility u rationalizes some SCR. In the next section, we ask the converse question: which SCRs can be rationalized by some utility function? We show the answer is almost allof them.

4. Knowing Costs Only: An Impossibility Result
In this section, we study the analyst’s ability to predict the agent’s behavior without knowing
anything about her utility. Thus, we take the agent’s cost function as given and ask which SCRs
can be optimal for some utility function, that is, which SCRs are rationalizable.
Clearly, the agent’s cost function places some restrictions on the set of rationalizable SCRs.
For instance, s can only be rationalizable if it is feasible; that is, only if it is in the set of SCRs thatare attainable at a finite cost,S κ := κ−1 (R).
The agent’s cost function can also rule out some SCRs whose costs are finite. For example, whenever Ω is finite and C equals mutual information (Example 1), only SCRs that use all actions intheir support in all states are rationalizable; that is, s is rationalizable only if sa is strictly positive
almost surely whenever it is not identical to zero (Caplin, Dean, and Leahy, 2019). Note, however,that this restriction has little practical relevance, because every SCR violating the restriction can
12
13

Observe E[u · s] is well-defined because ua and sa belong to the dual spaces L1 (µ0 ) and L∞ (µ0 ), respectively.
Compactness follows from the Banach-Aloaglu theorem (see, e.g., Theorem 6.21 in Aliprantis and Border, 2006).

10

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0
0

κ

κ

κ

0.6

0.2

0
0.2 0.4 0.6 0.8
s1 (y)

1 1

0

0.5

s1 (x)

00
0.2 0.4 0.6 0.8

1 1

0.5

0

0

0.2 0.4 0.6 0.8

s1 (x)s1 (y)s1 (y)

1 1

0.5

0

s1 (x)

Figure 1: Graphs of the indirect cost function κ and some of its subgradients for the case in which Ω ={x, y}, A = {0, 1}, and C is given by mutual information (see Example 1). The cost function is drawn as afunction of s1 (x) and s1 (y).

be approximated arbitrarily well by SCRs that satisfy it.
The above discussion raises the following question: Does C impose any meaningful restrictions
on the set of rationalizable SCRs beyond feasibility? The following result shows the answer isnegative. To state the result, call a set of SCRs S̃ uniformly dense in S κ if every s ∈ S κ and  > 0
admit some t in S̃ such that each a ∈ A and µ0 -almost every ω ∈ Ω have |sa (ω) − ta (ω)| ≤ .
Proposition 1. The set of rationalizable SCRs is uniformly dense in S κ .14
For intuition, recasting the agent’s maximization program in geometric terms is useful. To that
end, recall s is u-rationalizable if and only if
E[u · s] − κ(s) ≥ E[u · t] − κ(t) for all t ∈ S,
which is clearly equivalent toκ(t) ≥ κ(s) + E[u · (t − s)] for all t ∈ S.
The above condition has a geometric interpretation: s being u-rationalizable is equivalent to u
being a subgradient of κ at s. It follows s is rationalizable if and only if the set of all such subgradients()∂κ(s) := u ∈ U : κ(t) ≥ κ(s) + E[u · (t − s)] for all t ∈ S ,also known as κ’s subdifferential at s, is nonempty. The proposition then follows from observing
that κ is the convex conjugate of a continuous convex function, and so one can apply a dual versionof the Brøndsted-Rockafellar theorem (Brøndsted and Rockafellar, 1965).
The relationship between subgradients and optimality is a standard fact from convex analysis
14

If Ω is finite, every SCR in the relative interior of S κ is rationalizable.

11(see, e.g., Theorem 23.5 in Rockafellar, 1970).15 Geometrically, u is a subgradient of κ at s if
the function t 7→ κ(s) + E[u · (t − s)] defines a hyperplane whose graph supports the epigraphof κ at the point (s, κ(s)). Figure 1 visualizes this condition when Ω = {x, y}, A = {0, 1},
and C is given by mutual information. In this case, each s can be summarized by the probability
it takes action 1 in each state, (s1 (x), s1 (y)). The left panel illustrates κ as a function of theseprobabilities. When s1 (x) = s1 (y), the action generated by s is uninformative about Ω, and so theagent’s learning costs are minimized. Such SCRs are rationalizable by any utility that is constantacross both actions. Figure 1’s middle panel depicts the supporting hyperplane that results fromsuch utility functions. As can be seen, the resulting hyperplane lies everywhere below the graph ofκ, touching the graph only at the SCRs that are rationalizable via such utilities. For non-constantutilities, the agent usually finds it optimal to collect some information about the state. The figure’s
right panel depicts a supporting hyperplane that corresponds to such a situation.
Caplin, Dean, and Leahy (2021) prove a specialization of Proposition 1 for the case in which C
is posterior separable and when all SCRs in the interior of S have a finite cost. Their argument constructs a utility function that rationalizes an interior s by choosing ua to be an appropriate memberof c’s subdifferential at µsa . This construction suggests a connection between the subdifferentials
of c and κ whenever C happens to be posterior separable. In section 7, we discuss this connection
in greater detail.
One potential concern with Proposition 1 is that it could be driven by indifference. For an
extreme example, suppose learning is free; that is, C(p) = 0 for all p. In this case, only utility
functions that do not depend on the agent’s action (meaning ua = ua0 for all a, a0 ) can rationalize
a conditionally full-support SCR. Such indifference seems problematic, because it leaves open thepossibility of obtaining meaningful restriction on choices via the introduction of an appropriate
way to refine optimal behavior.
Next, we state some assumptions under which one can rationalize essentially every SCR without relying on indifference.
Assumption A1. .
(i) The cost function C is strictly monotone.
(ii) The cardinality ranking |Ω| ≥ |A| holds.
(iii) The domain S κ has a nonempty (norm) interior in S.
Part (i) means learning more always comes at a strictly positive cost. This part rules out indiffer15

This relationship is analogously used to connect additive random utility models (e.g., McFadden, 1974) with models of additive perturbed utility (e.g., Fudenberg, Iijima, and Strzalecki, 2015). In particular, Hofbauer and Sandholm(2002) and Norets and Takahashi (2013) use convex analysis to show every additive random utility model admits anadditive perturbed utility representation.

12

ence driven by some information being free. Part (ii) requires the set of states to be richer than theset of actions. Whenever part (ii) fails, the information policy ps associated with any full-support s
can be written as the convex combination of two other information policies, each of which is associated with a different SCR. Hence, if C is posterior separable (i.e., affine) and Assumption A1(ii)fails, s cannot possibly be uniquely rationalizable when s has full support.16 Assumption A1(iii)
plays a similar role to A1(ii). Without Assumption A1(iii), the cost C can restrict the agent to a
low-dimensional set of information policies, a restriction with similar implications to violations ofAssumption A1(ii).17
Our next result shows that, under Assumption A1, the inability to predict the agent’s behavior
using her information costs alone is not driven by indifference.
Theorem 1. Under Assumption A1, the set of uniquely rationalizable SCRs is uniformly dense inS κ .18
The above result is based on a particular set of SCRs that we call linearly independent. Formally, an s ∈ S is linearly independent if {sa }a∈A ⊆ L∞ (µ0 ) consists of |A| linearly independent
elements. In the appendix, we show such SCRs satisfy two useful properties. First, if s is linearly
independent and C is strictly monotone (i.e., Assumption A1(i) holds), then s is optimal only if it
is uniquely optimal.19 Second, the set of linearly independent SCRs is open and dense wheneverparts (ii) and (iii) of Assumption A1 both hold. Since the set of rationalizable SCRs is dense, andthe intersection of a dense set with an open and dense set is dense, we obtain that, under Assumption A1, the set of linearly independent SCRs that is rationalizable is dense, and that every such
SCR is, in fact, uniquely rationalizable.
In addition to showing learning costs alone impose no substantial restrictions on behavior, theabove analysis also highlights the futility of searching for cost functions that require the agent’s
behavior to satisfy certain desiderata. To illustrate, consider the following property, suggested byMorris and Yang (2021): a cost function C satisfies continuous choice if only continuous SCRs
are rationalizable.20 Morris and Yang (2021) suggest continuous choice as a way to model agents
who have a hard time distinguishing between similar states. Below, we show the only way to
16
In the appendix, we show that without Assumption A1(ii), a full support s can be uniquely rationalizable only if
C is not affine around ps .
17
Assumption A1(iii) holds, for example, if the agent can obtain a signal that reveals the state with probability ,and provides no information otherwise. It also holds for mutual information costs (Example 1).
18
We also show this set of SCRs admits a dense subset that is open when Ω is finite, and that the set of utilities thatrationalize an s in this subset is open (regardless of whether Ω is finite).
19
This fact generalizes results by Caplin and Dean (2013) and Matějka and McKay (2015) about uniqueness of an
optimum under mutual information costs, and Sharma, Tsakas, and Voorneveld’s (2020) Proposition 1(b) about uniquerationalizability under posterior separable costs with binary states. See the appendix for more details.
20
An SCR s is continuous if it admits a continuous version; that is, every a admits a continuous function fa : Ω →[0, 1] such that fa = sa almost surely.

13

guarantee this property is to make discontinuous information structures infeasible. To show thisresult, we observe that continuity of an SCR is a closed property, and so can be satisfied by a dense
subset of S κ only if it is satisfied by all of S κ ’s elements.
Proposition 2. Every rationalizable SCR is continuous if and only if every SCR in S κ is continuous.
Thus, our results suggest that except in cases where continuous choice is vacuous, it can only
hold if discontinuous SCRs are infeasible, a property referred to by Morris and Yang (2021) asinfeasible perfect discrimination.21 In terms of C, this property is equivalent to C assigning infinite
cost to every simple information policy that generates a µ with a discontinuous Radon-Nikodym
derivative with respect to µ0 . In section 7, we note one can circumvent the need for infinite costs byrequiring continuous choice to hold only for a restricted set of objectives. For example, one may
require the agent’s choice to be continuous for any bounded objective. As we explain later, sucha requirement is equivalent to a slight weakening of Morris and Yang’s (2021) expensive perfect
discrimination property.

5. Cross-Menu Predictions
The previous section shows learning costs alone impose no meaningful restrictions on the agent’s
behavior given a fixed menu. In this section, we highlight these costs do constrain how the agent
behaves across menus. The reason is that the agent’s choices from one menu allow the analyst
to use the agent’s cost function to extract information about her preferences. In fact, we show
the information revealed by the agent’s behavior is particularly sharp when her cost function issmooth in a manner we make precise below. This sharpness allows us to prove the following
result: whenever our smoothness conditions holds, most choices in one menu pin down the agent’s
actions in all submenus.

5.1. Smooth Costs
We introduce our smoothness condition in stages. We begin by defining a differentiability notion
for C. To that end, we recall some standard notation. Given a convex set X in a real vector space,a convex function f : X → R, and x ∈ f −1 (R), define the directional derivative of f at x asd+
x f : X → R ∪ {±∞} via
001d+x f (x ) := lim  [f (x + (x − x)) − f (x)] .
&0

21

We should highlight that we are abusing terminology slightly: Morris and Yang (2021) require Ω to be a realinterval, and say that a cost function satisfies infeasible perfect discrimination if it assigns finite cost only to SCRs thatare absolutely continuous.

14

Say c ∈ C is a derivative of C at p ∈ P C if c(µ) p(dµ) is finite and if every p0 ∈ P C hasR0d+p C(p ) =Zc(µ) (p0 − p)(dµ).

The cost function C is differentiable at p ∈ P C if it admits a derivative at p.22 We omit thedependency on the information policy and say C is differentiable whenever it is differentiable atall p ∈ P C .23
Intuitively, a cost function is differentiable if it prices small shifts in its information in a
posterior-separable manner. Indeed, observe that the function
Cc (q) =Zc(µ) q(dµ)

defines a posterior-separable cost function. In fact, this cost equals C whenever the latter is alreadyposterior separable: in this case, C has a common derivative at all information policies.
In general, the derivative of a differentiable cost function that is not posterior separable depends
R
on p. For a demonstration, consider the cost function in Example 4, C(p) = ψ ( c(µ) p(dµ)) .
Whenever ψ is differentiable, this cost function admits
ψ0

Z



c(µ) p(dµ) c(·)

as its derivative at p, which depends on p whenever ψ is not affine. For another example, consider
the cost function from Example 5. This cost function is differentiable, with a derivative at p ∈ P C
Rgiven by 2 c̃(·, µ) p(dµ). Clearly, this derivative typically changes as p varies.24
Our next result tightens the connection between posterior separability and differentiability. To
state this result, define the indirect cost function associated with Cc ,κc (t) := Cc (pt ).
Lemma 3 below shows that, whenever C has full domain, s is u-rationalizable if and only if it is
u-rationalizable when costs are given by the cost function’s posterior separable approximation at
ps .
22

Note that we require the derivative to be convex (because c ∈ C). In the appendix, we show one can omit thisrequirement whenever c is finite and continuous.
23
Our notion of differentiability is commonly used in the decision theory literature, where it is often called Gâteauxdifferentiability (e.g., Hong, Karni, and Safra, 1987; Cerreia-Vioglio, Maccheroni, and Marinacci, 2017). The definition is slightly different than the way Gâteaux differentiability is defined in convex analysis: whereas the latter requires
the convergence to occur from all possible directions (e.g., Phelps, 2009; Borwein and Vanderwerff, 2010), we requireconvergence only from directions within the domain of C.
24
For a non-differentiable cost function, consider Example 6 when C˜ is finite and not equivalent to a singleton. The
cost is not differentiabile at any p on the boundary between two regions where the set of maximizers differs.

15

Lemma 3. Fix some s ∈ S and u ∈ U. Suppose C is finite on P F and that c is a derivative of C
at ps .25 Then,s ∈ argmaxt∈S [E [u · t] − κ(t)]if and only ifs ∈ argmaxt∈S [E [u · t] − κc (t)] .
Note the above result does not tell us κ and κc have the same set of maximizers. The reason isthat the cost function’s derivative depends on the SCR around which the cost is approximated. In
other words, u-rationalizability of s is equivalent to s being u-rationalizable under Cc only if c isa derivative of C at ps . Unless C is posterior separable (in which case all SCRs admit a common
derivative), different SCRs usually admit different derivatives.
Next, we define a differentiability notion for derivatives of C. Given c ∈ C and a simply drawn
Rµ, we say ∇cµ ∈ L1 (µ0 ) is a derivative of c at µ if ∇cµ (ω) µ(dω) = c(µ), and every simply
drawn µ0 ∈ ∆Ω has
Z
0
d+
c(µ
)
=
∇cµ (ω) (µ0 − µ)(dω).
µ
The function c is differentiable at µ if it admits a derivative there. Thus, c is differentiable if it canbe locally approximated by an affine function.
Our key notion of smoothness requires C to admit a differentiable derivative. Specifically, we
say C is iteratively differentiable at p ∈ P F if it admits a derivative c at p that is differentiable atevery µ ∈ supp p. Thus, an iteratively differentiable cost is locally similar to a smooth, posteriorseparable cost function.
The benefit of having iteratively differentiable costs is summarized in the following proposition, which tightly characterizes when an interior SCR is optimal.
Proposition 3. Suppose s has full support, and C is finite on P F and iteratively differentiable at
ps with derivative c. The following are equivalent for u ∈ U:(i) SCR s is u-rationalizable; that is, s ∈ argmaxt∈S [E [u · t] − κ(t)].
A(ii) Some λ ∈ L1 (µ0 ) and γ ∈ L1 (µ0 )+ exist such that every a ∈ A haveua = λ − γa + ∇cµsa and γa sa = 0.
One can view the proposition as establishing a Lagrange multiplier result for iteratively differentiable costs, with ∇cµsa serving the role of the derivative of κ. To better see this interpretation,
One can weaken the requirement that C is finite on P F , assuming only that C is finite for simple information
policies around ps . See the appendix for more details. Proposition 3 below can be similarly strengthened.
25

16

suppose Ω is finite and that κ is differentiable. In this case, s being u-rationalizable is equivalent
to s solving the program
max
A×Ω

h

t∈R

s.t.

i

E [u · t] − κ(t)ta (ω) ≥ 0∀a ∈ A, ∀ω ∈ Ω,X∀ω ∈ Ω.

ta (ω) = 1

aApplying a standard Lagrangian result (e.g., Pourciau, 1983) gives a multiplier γa (ω) for every
instance of the first constraint and a multiplier λ(ω) for every instance of the second constraint
such that s is optimal if and only if
ua (ω) −∂κ
1+ γa (ω) − λ(ω) = 0.
µ0 (ω) ∂sa (ω)

Observe the above display equation specializes to condition (ii) of the proposition, but with µ01(ω) ∂s∂κa (ω)
replacing ∇cµsa .
In addition to characterizing optimality of a full-support s, Proposition 3 also enables one torecover the agent’s utility function from their behavior, up to a nuisance term. For an explanation,
suppose s has conditionally full support and that C is iteratively differentiable at ps with derivativec. Let us be the utility function defined via usa := ∇cµsa for all a. Then, Proposition 3 implies
a utility u rationalizes s if and only if u generates the same optima for all menus. The reason isthat u equals us plus an action-independent shift, λ, the addition of which has no impact on theset of maximizers.26 The next section uses this observation to identify conditions under which theagent’s choices in one menu are sufficient for pinning down her behavior for all submenus.

5.2. Unique Subset Predictions
We now use the results of the previous section to show the agent’s learning costs restrict her behavior across menus. To model these restrictions, let A be all nonempty subsets of A. For any
P
B ∈ A, let SB = {s ∈ S : a∈B sa = 1} be the set of SCRs that only use actions in B. Given autility function u, we say s ∈ SB is u-rationalizable over B if s solves the agent’s problem when
the agent is restricted to only using actions in B; that is,s ∈ argmaxt∈SB [E [u · t] − κ(t)] .
26

Note γa sa = 0 means γa = 0 when s has conditionally full support.

17

Say s ∈ S yields unique subset predictions if every B ∈ A has a unique t ∈ SB that admits autility that both rationalizes t over B and rationalizes s (over A). Thus, if the agent chooses from
A according to s, and s yields unique subset predictions, we can deduce exactly what must be theagent’s chosen SCR when she’s restricted to B.
Next, we introduce assumptions that enable us to conclude many SCRs yield unique subset
predictions. To state the assumption, say an information policy is fully mixed if all posteriors in
its support have a strictly positive Radon-Nikodym derivative with respect to µ0 .
Assumption A2. .
(i) C is finite at every simple information policy.27(ii) C is iteratively differentiable at every simple and fully mixed information policy.
(iii) If p ∈ P F is not fully mixed, d+
p C(δµ0 ) = −∞.
The first two parts of the assumption require C to be smooth, as explained in the previous
section. The third part requires C to have infinite slopes at simple information policies that are notfully mixed. This property, which is satisfied by mutual-information costs (Example 1), implies an
SCR can be rationalizable only if it uses all actions in its support in all states. In particular, a fullsupport SCR is rationalizable only if it has conditionally full support. Therefore, Assumption A2
implies one can use Proposition 3 to recover the agent’s utility function, up to a choice-irrelevant
nuisance term, whenever she employs a full-support SCR. As Lemma 4 below shows, the agent’s
utility function enables the analyst to precisely predict the agent’s conditional action distribution,
except for a knige-edge set of utilities.
Lemma 4. The set of utilities with multiple rationalizable SCRs is meager and shy.28
Lemma 4 essentially follows from the fact that most hyperplanes that support the graph of awell-behaved convex function admit a unique support point. Figure 2 illustrates a sense in which
this fact holds for a convex function φ defined over a closed subinterval of R. In one dimension, a
subgradient is defined via the slope of the corresponding line. For a line to support φ in multiple
points, the line’s slope must equal the slope of φ in an interval over which φ is affine. Clearly,
at most countably many such intervals can exist, and so the set of gradients that admit multiple
support points must be countable as well. Being countable, the set is also meager and Lebesgue
null, two properties that generalize to higher dimensions.
This condition implies S κ = S, and, in particular, implies A1(iii).
A set is meager if it is a countable union of nowhere dense sets. A set V ⊆ U is shy if some probability measureν ∈ ∆U with compact support assigns zero measure to every translation of V, that is, if ν(V + u) = 0 for all u ∈ U.
Shy sets generalize Lebesgue null sets beyond the case of finite dimensions (see Hunt, Sauer, and Yorke, 1992). Thus,
the proposition implies that for finite Ω, the set of utilities that admit multiple rationalizable SCRs is Lebesgue null.
27

28

18

Figure 2: A convex function over an interval with the set of its subgradients that attain multiple support
points.

Using the above facts, our next result demonstrates that the agent’s learning costs often impose
strong restrictions on the agent’s behavior across menus.
Theorem 2. Under A1 and A2, SCRs yielding unique subset predictions are weak* dense.
To prove the theorem, we first use Lemma 4 to find a dense set UA of utilities that attain aunique optimum at every menu. The proof then proceeds by identifying a dense subset of SCRs
that are rationalizable by utilities in UA . To identify this subset, we observe Theorem 1 and Assumption A2(iii) imply the set of uniquely rationalizable SCRs with conditionally full support isdense in S κ . Using a continuity property of the subdifferential, we then show one can approximate
any s in this set with SCRs that are rationalizable by some utility in UA . Moreover, because s
has conditional full support, the approximating SCRs can be taken to have the same property.29
Focusing on one of the approximating SCRs, Proposition 3 implies all utilities that rationalize it
generate the same optima for all menus. Because one of these rationalizing utilities comes from
UA , it follows that, for any given menu, all of these utilities uniquely rationalize the same SCR.
In other words, each of the SCRs approximating s yields unique subset predictions. Since s is anarbitrary element of a dense subset of S κ , the result follows.

6. Cross-Menu Tests
In previous sections we explored the predictive content of the agent’s costs under two extreme
scenarios: section 4 considers the case of no auxiliary data, whereas section 5 studies the casewhere this data is ideal. In this section we explore the middle ground, studying the restrictions
imposed by the agent’s cost function in an arbitrary data set. In doing so, we develop a test forwhether the agents’ choices are consistent with a given cost function.
Throughout this section, we focus on cost functions that satisfy Assumption A2(i) and A2(ii),which are satisfied by virtually all models used in the literature.
29

This property of the approximating SCRs relies on Assumption A2(iii). We note this assumption is not needed
when Ω is finite, because in this case, the set of conditionally full-support SCRs is open.

19

We consider an analyst who can estimate the agent’s chosen SCRs in a subset of all possible
menus. Specifically, we define a data set to be a pair, D = (B, β), where the set B ⊆ A contains
the menus on which the analyst can estimate the agent’s behavior, and β : B → S maps each suchmenu to the SCR describing the agent’s behavior in that menu, and so has β B := β(B) ∈ SB foreach B ∈ B. Such data sets have been produced in the literature, see Dean and Neligh (2022), andCaplin et al. (2020), for example. Our main goal is to develop necessary and sufficient tests for adata set to be compatible with a given cost function and a common objective. Specifically, we areinterested in testing whether a data set D is consistent, meaning a utility function u ∈ U exists suchthat, for every B ∈ B, the SCR β B is u-rationalizable over B. We find it convenient to concentrateon data sets in which the agent takes all feasible actions with positive probability conditional on thestate. These are data sets that have conditionally full support, namely βaB (ω) > 0 holds almost
surely for all a ∈ B ∈ B. We discuss data sets without this property at the end of this section.
Our tests are based on a particular class of cycles generated by the data. To describe thesecycles, note that every data set induces a bipartite graph GD with vertex classes B and A, and forwhich an edge exists between B ∈ B and a ∈ A if and only if a ∈ supp β B . A testable cycle
is a cycle in this graph that begins (and therefore ends) with an action. In other words, a testable
cycle is a sequence a0 B1 a1 B2 a2 . . . BN aN such that a0 = aN , and {an−1 , an } ⊆ supp β Bn for alln ∈ {1, . . . , N }.
To better understand the concept of testable cycles, consider the following example. The
agent has three actions A = {x, y, z}, the utility of which depends on a state that is uniformly
distributed over {1, 2, 3}. The analyst can estimate the agent’s chosen SCRs on three menus,
B = {{x, y, z}, {x, y}, {y, z}}, with the agent’s behavior in these menus is given by







1 2 2


,
,
 5 5 5




, ,
5 5 5






 2, 2, 1

if ω = 2,

if ω = 1,





1 2


,
,
0


 3 3 

, ,0
3 3






 1, 1, 0

if ω = 2,

if ω = 1,





1 1


0,
,


 2 2 

βx{x,y,z} , βy{x,y,z} , βz{x,y,z} (ω) =

2 1 2

5 5 5



βx{x,y} , βy{x,y} , βz{x,y} (ω) =

2 1

2 2



βx{y,z} , βy{y,z} , βz{y,z} (ω) =1 2

0, 3 , 3






 0, 2 , 1
3 3

if ω = 1,

if ω = 3,(2)

if ω = 3,

if ω = 2,
if ω = 3.

In this example, the data set β has conditionally full support. Therefore, the bipartite graph induced

20

x{x, y}

x{x, y}

x{x, y}

x{x, y}

y{x, y, z}y{x, y, z}y{x, y, z}y{x, y, z}z{y, z}z{y, z}z{y, z}z{y, z}

Figure 3: The left panel depicts the bipartite graph induced by a conditionally full support data set (B, β)with B = {{x, y}, {x, y, z}, {y, z}}. The right three panels depict the three simple cycles contained in thisgraph.

by β has an edge between an action a and a menu B ∈ B if and only if a ∈ B. In other words,
this data set induces the graph drawn in Figure 3. This graph contains three simple testable cycles:
x {x, y} y {x, y, z} x, y {x, y, z} z {y, z} y, and x {x, y} y {y, z} z {x, y, z} x. Every othertestable cycle involves concatenations of these three cycles (possibly in reversed order or startingat a different action).
It turns out to be convenient to focus on a special subset of testable cycles called a cycle basis.
To define such a subset, let ED be the set of all edges in the graph induced by D—that is, theset of all pairs, {a, B} such that B ∈ B and a ∈ supp β B . Associate each testable cycle χ =a0 B1 a1 B2 a2 . . . BN aN with the cycle vector `χ ∈ RED given by`χ{a,B} = |{n : (a, B) = (an−1 , Bn )}| − |{n : (B, a) = (Bn , an )}| .
Let L be the vector subspace of RED spanned by the set of all cycle vectors. A cycle basis is a setof cycles whose corresponding vectors form a basis for L. Informally, a cycle basis is a minimal
set of cycles that is sufficient for generating any other cycle in the graph. For more information
on different types of cycle basis and algorithms that generate them, see a standard reference, e.g.,
Bollobás (2012).30 For the current purpose, it suffices to note that several algorithms exist forfinding a cycle basis, and that one can easily calculate a basis’ cardinality: any cycle basis of thegraph induced by a given data set D contains
X|supp β B | − |A| − |B| + k

(3)

B∈B

cycles, where k is the number of connected components of the graph.
To demonstrate the definition of cycle basis, consider again the data set presented in equation (2). As noted earlier, every testable cycle in this example is essentially a concatenation ofsome combination of the following three simple cycles: χ1 = x {x, y} y {x, y, z} x, χ2 =y {x, y, z} z {y, z} y, and χ3 = x {x, y} y {y, z} z {x, y, z} x. However, the calculation from
30

Bollobás (2012) Chapter II, §3 defines cycle bases, characterizes their cardinality, and provides an explicit construction of one.

21

equation (3) says that only 2 cycles are necessary for obtaining a cycle basis. We now explain that{χ1 , χ2 } is a cycle basis. To do so, observe first that the corresponding set of vectors {`χ1 , `χ2 } islinearly independent, because each cycle contains an edge that is not included in the other. Second,
recall every cycle is built from the three previously-mentioned cycles. Therefore, for every cycle
χ, one can find three integers k1 , k2 , k3 ∈ Z such that `χ = k1 `χ1 + k2 `χ2 + k3 `χ3 . And third,observe `χ3 = `χ1 − `χ2 . It follows {`χ1 , `χ2 } is a linearly independent set of vectors that spans
L—i.e., {`χ1 , `χ2 } is a cycle basis.
Testable cycles are important because they trace all the restrictions imposed by the consistency
requirement. To introduce these restrictions, we require some additional notation. Fix some dataB
βB
set D = (B, β). For each B ∈ B, let cB be some derivative of C at pβ , and take µB
a = µa to beposterior revealed by action a given SCR β B . Then for every B and action a ∈ supp β B , we candefine the function
Dfa,B:= ∇cB.
µB
aArmed with these definitions, we can associate each testable cycle a0 B1 a1 B2 a2 . . . BN aN with thefollowing equation31
N 
X



faDn ,Bn − faDn−1 ,Bn = 0.

(4)

n=1

Corollary 1 below states that a conditionally full-support data set is consistent only if equation (4)holds for all testable cycles. This corollary also shows that any conditionally full-support data set
satisfying equation (4) for all testable cycles is consistent. In fact, the corollary shows somethingstronger: to prove that a data set is consistent, it suffices to establish equation (4) only for thetestable cycles contained in a given cycle basis.
Corollary 1. Suppose C satisfies Assumptions A2(i) and A2(ii), and let D be a data set withconditionally full support. Given any cycle basis, the following are equivalent:(i) The data set D is consistent.
(ii) Equation (4) holds for every testable cycle.
(iii) Equation (4) holds for every testable cycle in the cycle basis.
We now briefly discuss the logic behind Corollary 1. The equivalence between (ii) and (iii)follows from observing that equation (4) requires all cycle vectors to be in the kernel of a fixedlinear map. Thus, the crux of the corollary is in the equivalence between parts (i) and (ii). To
understand this equivalence, revisiting Proposition 3 is useful. According to this proposition, aD
Although the definition of (fa,B)a∈B depends on the choice of derivative cB for each B, this dependence shiftsDfa,Bin the same way for every a ∈ B. Because each testable cycle has exactly one outgoing edge from B for each
incoming one, the left-hand side of equation (4) turns out not to depend on this choice of derivative.
31

22

conditionally full support data set D = (B, β) is consistent if and only if we can find two vectorsof L1 (µ0 ) functions, (ua )a∈A , and (λB )B∈B , such that for all a ∈ A and B ∈ B,D
= ua − λB .
fa,B

(5)

Thus, Proposition 3 reduces the question of whether a data set is consistent to a specific instanceof a standard problem in electrical networks: when can one assign each a ∈ A an electric potentialDua , and each B ∈ B an electric potential λB , so that fa,B
gives the difference in potentials between
a and B? The answer is that such an assignment is feasible if and only if f D satisfies Kirchhoff’s
voltage law, which corresponds exactly with equation (4) holding for all testable cycles. Thus,
Corollary 1 follows directly from Proposition 3 and the logic behind Kirchoff’s law.
As noted above, Part (iii) of Corollary 1 tells us that to establish consistency of a data set, it
suffices to test equation (4) only for the cycles contained in some cycle basis. One might wonder
whether it is possible to do the same with a different, potentially smaller set of cycles. The next
result establishes the answer is negative.
Proposition 4. Suppose C satisfies Assumptions A1 and A2. Let B ⊆ A be any set, and let X bea set of testable cycles (for full-support data sets on B) that contains no cycle basis. Then someinconsistent, conditionally full-support data set (B, β) exists such that equation (4) holds for every
cycle in X .
The above results allow analysts to test for a particular cost function without making assumptions on the agent’s payoffs. Consider the task of testing whether costs are given by mutual information, for example. The experimental literature focuses on the Independent Likelihood Ratio(ILR) condition (see Caplin and Dean, 2013; Dean and Neligh, 2022), which states that for finite
R
states, s is optimal when costs are given by C(p) = θ K(µ) p(dµ) for some θ > 0 only ifµsb (ω)µsa (ω)=eua (ω)/θ
eub (ω)/θ
for all a and b that s generates with positive probability. Testing this condition requires analysts
to know the agent’s payoffs from different actions. To pin down these payoffs, experimenterseither assume participants’ payoffs are linear in money (see Caplin and Dean, 2013), or craft theirexperiment so that rewards are given via “probability points” (e.g., Caplin et al. (2020), or Dean
and Neligh (2022)). By contrast, Corollary 1 allows one to test for mutual information costs using
the agent’s decisions across menus. Specifically, Corollary 1 implies a data set with conditionallyR
full-support β is consistent with the cost function C(p) = θ K(µ) p(dµ) if and only if every

23

testable cycle satisfies equation (4), with32
βaB
Dfa,B = θ log β B .
pa
For a demonstration, consider again the data set from equation (2). As explained earlier, the twocycles χ1 = x {x, y} y {x, y, z} x and χ2 = y {x, y, z} z {y, z} y form a cycle basis. Hence,
checking whether equation (4) holds for these two cycles is sufficient for testing consistency ofthat data set. Since there are three states, each of these cycles generates three equations, giving a
total of six equations to check. For example, evaluating equation (4) for the first cycle χ1 at stateω = 1 givesDDDDfx,{x,y}(1) − fy,{x,y}(1) + fy,{x,y,z}(1) − fx,{x,y,z}
(1)

"

2/3
2/5
1/5
1/3
− log
+ log
− log
=θ log
1/2
1/2
1/3
1/3

#

=0.
More generally, one can show all instances of equation (4) holds for both cycles. Hence, one
can use Corollary 1 to show that the data set from (4) is consistent with the agent having mutual
information costs without having any payoff information.33
Corollary 1 is also useful for testing for cost functions that do not have a revealed preference
axiomatization. For an example, suppose one is interested in testing whether costs are given bymutual information to the power of θ > 1,
θZC(p) =

K(µ) p(dµ) .

D
In this case, the function fa,Bin equation (4) becomes

h

D
fa,B
= θ κM I (β B )

hP

iθ−1

log

βaB
pβa

B

,

i

where κM I (s) = E a∈A sa log psas is the indirect cost function for the mutual information case.
a
We now use our test to show the example from equation (2) is inconsistent with the cost C(p) =R[ K(µ) p(dµ)]θ any θ > 1. To see this, evaluate the left hand side of equation (4) for the cycle
32

To obtain this condition, observe that if s has full support, the iterated derivative of mutual information costs isdµs
given by ∇cµas = log dµa0 = log psas .
a33
Observe this test is independent of θ. This independence comes from the fact that an s is u-rationalizable whencosts are given by θC if and only if it is (u/θ)-rationalizable when costs are given by C.

24χ1 = x {x, y} y {x, y, z} x at state ω = 1 to get
D
D
D
D
(1) =
(1) − fx,{x,y,z}(1) + fy,{x,y,z}(1) − fy,{x,y}fx,{x,y}hκM I (β{x,y,z}

iθ−1

)

h

− κM I (β

{x,y}

)

iθ−1 

θ log 2,

which differs from zero for all θ > 1, because κM I (β {x,y,z} ) , κM I (β {x,y} ) and κM I ≥ 0. Thus,
this example data set is inconsistent with a strictly convex power of mutual information.
So far, we focused on data sets with conditionally full support. We conclude the section by
discussing what happens when this assumption is violated. For this purpose, we introduce two
definitions. First, say a data set D = (B, β) is fully mixed if βaB is strictly positive almost surely
whenever βaB is not identical to zero. In other words, a fully-mixed data set has SCRs that use
all actions in their support in all states, but may not use all available actions in some menus.
Second, refer to D = (B, β) as having full support if it always uses all available actions; that is, if
supp β B = B for all B ∈ B. Observe that a conditionally full-support data set is exactly one thatis fully mixed and has full support.
For a fully-mixed data set without full support, the restrictions imposed by (4) are necessary
for consistency, but not sufficient. The lack of sufficiency arises because (4) does not incorporate
the fact that certain actions are not taken in some menus. In Online Appendix C, we present an
example of a data set that satisfies equation (4) for all cycles but is inconsistent.
For full-support data sets that are not fully mixed, our test is sufficient for consistency, but is notnecessary. For intuition let us return to Proposition 3. This proposition implies that a full supportB
data set is consistent if and only if we can find a utility u ∈ U, and multipliers λ ∈ L1 (µ0 ) andA×B
γ ∈ L1 (µ0 )+ , such thatD∀a ∈ A and B ∈ B : fa,B= ua − λB + γa,B and γa,B (·)βaB (·) = 0 almost surely.

(6)

As explained earlier, our test is equivalent to satisfying the above with the additional requirementthat γ is zero. Thus, our test is clearly sufficient for consistency of a full-support data set. Whenthe data set is also fully-mixed, our test is also necessary, because γa,B (·)βaB (·) can be zero for alla ∈ A and B ∈ B if and only if γ = 0. However, without the fully-mixed requirement, our test is
too strong, because γ need not equal zero for (6) to hold. And indeed, Online Appendix C presents
an example of a consistent data set that fails the test outlined in Corollary 1.34
34

Observe that any such example must violate Assumption A2. Indeed, if Assumption A2(iii) were satisfied, every
consistent full-support data set would have conditionally full support.

25

7. Discussion
In this section, we discuss our model’s assumptions, additional results, and the relationship toexisting literature.
Partial Knowledge of Benefits. For a fixed menu, we studied the restrictions imposed on theagent’s behavior by her information acquisition costs with complete knowledge and complete ignorance of her preferences. In the appendix, we develop tools for analyzing an intermediate casein which the analyst knows the agent’s utility belongs to a well-behaved set, V ⊆ U. In particular,
we characterize when an SCR s can be rationalized by some utility in V. The characterization
involves comparing the directional derivative of κ at s with the support function of V. Intuitively,
the directional derivative gives the marginal cost of shifting behavior away from s, whereas the
support function gives the maximum benefit of shifting one’s SCR towards s. Our result showsthat, whenever the latter is lower than the former, some utility in V rationalizes s.
To apply the above-mentioned result, one needs to calculate the support function of the set ofutilities V, and the directional derivative of the agent’s indirect cost κ. Calculating the support
function of V is a standard optimization problem. In the appendix we solve this problem explicitly
for a few economically relevant sets of utilities.
In general, calculating the directional derivative of κ can be a complicated problem. To simplify
it, we show that for differentiable costs, one can replace κ with its posterior separable approximation, κc . This replacement is useful, because one can calculate the directional derivative of κc using
the directional derivative of c, which is often easier to derive. We refer the reader to the appendix
for the exact statement of these results.
Partial Knowledge of Costs. Our model assumes the analyst knows the agent’s learning costs
exactly. Maintaining this stylized assumption enables us to study the degree to which the agent’s
learning costs pin down her behavior. In practice, many analysts may not know C exactly, but are
instead capable of restricting it to belong to some set C. Some of our results also speak to this case.
For example, Proposition 1 implies the set of SCRs that is rationalizable by some cost function in
C is uniformly dense in the set of SCRs that are feasible for some C ∈ C. Similarly, if each C ∈ C
satisfies A1, Theorem 1 immediately implies the set of uniquely rationalizable SCRs is uniformly
dense in the set of SCRs that can be induced at finite C-cost for some C ∈ C. Lemma 4 also extends
somewhat: if C is countable, the set of utilities that does not generate a unique prediction for someC ∈ C is meager and shy. Hence, for most utility functions, the analyst’s uncertainty about theagent’s behavior reduces to the uncertainty about C—provided C is countable. The reason is thata countable union of meager and shy sets is itself meager and shy. By contrast, we do not know of26

an immediate way to extend Theorem 2 to accommodate multiple cost functions.
Generalizing Corollary 1 to a set C of cost functions satisfying Assumption A2(i) and (ii) is
straightforward: a data set is consistent with the set C if and only if the Corollary 1’s test holds forsome cost function in the set. Even if the data set is consistent with C, one can still use Corollary 1’s
test to identify which subset of C could have generated a given data set. For an example, consider ananalyst equipped with the data set from (2) who believes costs are given by the LLR cost function(see Example 2), but is unsure about the value of the parameter θ. Using the test in Corollary 1,
the analyst can deduce θ must be symmetric. Moreover, every symmetric θ is consistent with thedata set in (2) (see Appendix B.2 for more details).
Unique Rationalizability and Strict Convexity. Our analysis showed that, under A1, one can ruleout indifference as the source of the analyst’s inability to predict behavior using the agent’s learning
costs. We focused on A1 because it accommodates the important case of posterior separable costs.
An alternative way to obtain a similar result is to look at costs that are strictly convex. We now
provide such a result.
Proposition 5. Suppose C is strictly convex on P C , and P C , {δµ0 }. Then, a set of uniquely
rationalizable SCRs exists that is uniformly dense in S κ and is open if Ω is finite. Moreover, thisset of SCRs is rationalized by an open set of utilities.
As the above result highlights, some rationalizable SCRs need not be uniquely rationalizable
even when C is strictly convex. The reason is that some convex combinations of SCRs change
the way the agent randomizes over actions conditional on her information without changing theinformation itself. For example, suppose the action set is binary, A = {0, 1}, and take s and t
to be the SCRs that respectively take action 1 and 0 regardless of the state. Clearly, both SCRs
reveal an uninformative information policy, ps = pt = δµ0 . Moreover, the same is true for any
convex combination of s and t, because any such combination results in the agent’s actions being
independent of the state. It follows the cost of any such convex combination is identical to the costof s and t. In other words, even when C is strictly convex, κ is still affine over some line segments.
To prove Proposition 5, we use strict convexity of C to identify regions where κ is strictly
convex. Specifically, we show κ is strictly convex over any line segment with an end point thatsatisfies the following property: every action is used with positive probability, and no two actions
reveal the same posterior. Therefore, any rationalizable SCR with this property is uniquely rationalizable. The proposition’s proof then proceeds as the proof of Theorem 1, but with SCRs withthe previously mentioned property taking the role of the set of linearly independent SCRs.
Next, we note substituting strict convexity of C for A1 does not alter the conclusions of Theorem 2.
27

Proposition 6. If C is strictly convex on its domain, |Ω| > 1, and A2 holds, the set of SCRs thatyield unique subset predictions is weak* dense.
The argument for Proposition 6 follows the same reasoning as Theorem 2. We refer the reader
to the appendix for the specific details.
Subdifferentials, Rationalizability, and Posterior Separable Costs. Among their many contributions, Caplin, Dean, and Leahy (2021) also identify a connection between rationalizability andsubdifferentiability for the case in which C is posterior separable. More specifically, Caplin, Dean,and Leahy (2021) show that if Ω is finite and C is posterior separable, s is rationalizable if and only
if c has a nonempty subdifferential at all beliefs in the support of s’s revealed information policy;
that is, ∂c(µsa ) , ∅ for all a ∈ supp s.35 Because rationalizability of s is equivalent to ∂κ(s) being
nonempty, Caplin, Dean, and Leahy’s (2021) result delivers the following conclusion: whenever Ω
is finite and C is posterior separable, ∂κ(s) is nonempty if and only if ∂c(µsa ) is nonempty for alla ∈ supp s.
One can decompose Caplin, Dean, and Leahy’s (2021) argument into two. First, they show one
can construct a utility function that rationalizes s by setting ua to be an appropriately normalized
member of ∂c(µsa ). In the appendix, we show Caplin, Dean, and Leahy’s (2021) construction
extends to infinite states and the case in which costs are merely differentiable. In other words, weshow ∂κ(s) is nonempty whenever C admits a derivative c at ps for which ∂c(µsa ) is nonempty for
all a ∈ supp(s).
Caplin, Dean, and Leahy (2021) also establish a converse: when costs are posterior separable
and the state is finite, s is rationalizable only if ∂c(µsa ) is nonempty for all a in s’s support. To
prove this claim, Caplin, Dean, and Leahy (2021) prove a duality result to obtain a Kuhn-Tuckerlike necessary condition for s to be u-rationalizable, and show adding the relevant multiplier to ua
witnesses ∂c(µsa ) being nonempty. Our results imply this approach generalizes to differentiable
costs as well. The reason is that, under Lemma 3’s conditions, s is rationalizable if and only if it
is rationalizable by C’s posterior separable approximation at ps . Thus, given s and a finite-valued
cost that admits a derivative c at ps , the SCR s is rationalizable only if ∂c(µsa ) is nonempty at alla ∈ supp(s)—provided the state is finite. Finite states are necessary because Caplin, Dean, andLeahy’s (2021) duality result may not apply with infinite states. To establish similar duality results
for the infinite state case, one usually needs additional regularity conditions (see, e.g., Gretsky,
Ostroy, and Zame, 2002; Dworczak and Kolotilin, 2019). Lacking such a result or an alternative
proof method, we do not know whether ∂κ(s) being nonempty implies the nonemptiness of ∂c(µsa )for all a ∈ supp(s) when the state is infinite.
35

See the definition of ∂c(µsa ) in Appendix A.5.

28

The construction from the first part of Caplin, Dean, and Leahy’s (2021) argument delivers
an alternative way of proving Proposition 1 for the case in which the state is finite and costs areposterior separable. Given our above-mentioned result, the same argument extends to costs thatare differentiable. For an explanation, recall the subdifferential of a convex function is nonempty
over the relative interior of its domain (see, e.g., Rockafellar, 1970, Theorem 23.4), which isalways nonempty in finite dimensions. Because an interior s reveals only posteriors that have full
support, and because all such posteriors are interior when the state is finite, one gets that, when Ωis finite, ∂c(µsa ) , ∅ for all a whenever s is interior. Therefore, when costs are differentiable, one
can use Caplin, Dean, and Leahy’s (2021) construction to prove Proposition 1 for the finite-statecase. However, with infinite states, Caplin, Dean, and Leahy’s (2021) approach does not deliver
an immediate proof for Proposition 1. The reason is that the relative interior of c’s domain isempty, and so ∂c(µsa ) may be empty as well. By focusing on the subdifferential of κ (which is welldefined even when C is not differentiable), our proof not only avoids this issue, but also establishes
Proposition 1 for a more general class of cost functions.
Convexity and Monotonicity. In our analysis, we assumed C is monotone and convex. We now
explain these two assumptions are essentially without loss. In particular, we argue the indirect costfunction generated by every lower semicontinuous and proper Ĉ is identical to the indirect cost
generated by a convex and monotone cost function.
To get the result, we must first redefine the agent’s indirect cost function so as to allow randomization over information policies. Specifically, we let the agent choose a distribution over
information policies, Q ∈ ∆P. We say such a distribution can induce an SCR s if some action
strategy α : ∆Ω → ∆A is such that, for every a ∈ A and every event Ω̃,Zα(a|µ)µ(Ω̂) p(dµ) Q(dp) = E [1Ω̃ sa ] .

The indirect cost function is then given byκ(s) = inf

Q∈∆P

Z

Ĉ(p) Q(dp) s.t. Q can induce s.

(7)Note randomization is not necessary when C is convex, in which case the above reduces to ourprevious definition of κ. The next result shows a sense in which such convexity always holds.
Proposition 7. Let κ be the indirect cost function induced by a lower semicontinuous and properĈ : P → R. Then, the infimum in (7) is attained. Moreover, κ is also induced by some costfunction C that is lower semicontinuous, proper, convex, and monotone.
Our argument begins by observing Q can induce s if and only if its mean, q =29Rp Q(dp),is more informative than s’s revealed information policy ps . It follows the agent’s indirect costfunction remains unchanged if we replace Ĉ with a cost function C that assign each p with the
expected cost (under Ĉ) of the cheapest randomization whose mean is more informative than p.
We then show Berge’s theorem guarantees C is lower semicontinuous and that monotonicity andconvexity of C follow from  being a transitive order that respects convex combinations.
Caplin and Dean (2015) use a similar construction to show every behavior generated fromcostly flexible learning in a finite collection of menus can be rationalized using a convex andmonotone cost function. de Oliveira et al. (2017) prove a representation result for preferences
over menus with similar implications. In particular, they show one can always take the cost of p
to equal the maximum difference between the agent’s benefit from using p in some menu and her
certainty equivalent for that menu. Moreover, the resulting cost function is the unique cost functionthat is simultaneously convex, monotone, zero at no information, and consistent with the agent’s
preferences over menus.
Continuous Choice with Bounded Utilities. Proposition 2 shows the only way to guarantee continuous choice across all objective functions is to require all discontinuous SCRs to have infinite
cost. We now explain one can avoid the use of infinite costs if one is willing to require the agent’s
choice to be continuous for all bounded utility functions. In particular, C generates continuous
choice for all bounded utility functions if and only if it satisfies an infinite-slope condition.
Proposition 8. An SCR s is not rationalizable by any bounded utility function if and only if
κ(s) − κ(t)= −∞.
t∈S \{s} ks − tk1
inf
κ

(8)

In particular, only continuous SCRs are rationalizable by a bounded utility function if and only if(8) holds for all discontinuous s.
The result is an immediate consequence of Gale (1967), who shows bounded steepness is anecessary and sufficient condition for the subdifferential of a convex function to contain somelinear function that is continuous with respect to a given norm.36
Proposition 8’s infinite-slope condition is reminiscent of a different condition by Morris andYang (2021), who introduce the notion of continuous choice to study equilibrium selection in
global games. Holding other players’ strategies fixed, one can view the problem of each agent intheir game as an instance of our model in which Ω ⊆ R is an interval, A = {0, 1}, payoffs arebounded, and S κ is contained in the space of all SCRs for which s1 is nondecreasing. Morris and36

Using identical reasoning, one can replace k · k1 in Proposition 8’s statement to analogously characterize which
SCRs are rationalizable by other subspaces of utilities. See Online Appendix B.6 for details.

30

Yang (2021) show multiplying κ by a vanishing constant leads to a sharp equilibrium-selection
result, provided only SCRs in SAC := {s ∈ S : s1 absolutely continuous} can be rationalized bysome monotone and bounded utility function. They also prove this latter property holds wheneverκ satisfies a condition called expensive perfect discrimination, which states that for every s ∈S κ \ SAC , the cost function κ exhibits unbounded k · k1 -steepness in the direction of SCRs in SAC .
By contrast, Proposition 8’s condition allows κ to exhibit unbounded steepness from any direction,and shows allowing for these additional directions results in a condition that is both necessary andsufficient for ruling out discontinuous SCRs as rationalizable by any bounded utility, including
utilities that are not monotone.
Costly Stochastic Choice. In our model, we assumed the agent faces a cost to acquire information, which we then used to derive an indirect cost function over the set of SCRs. By contrast,some models formulate a cost function κ̃ on SCRs directly, without micro-founding it via information acquisition (e.g., Mattsson and Weibull, 2002; Fosgerau et al., 2020; Flynn and Sastry, 2021;
Morris and Yang, 2021). Some of our results apply to those models as well, provided κ̃ is convex,
proper, and weak* lower semicontinuous. Because the arguments for Lemma 4 and Proposition 1
rely only on properties of the agent’s indirect cost function, both of these results also apply tomodels in which the cost of an SCR does not originate from information acquisition. The same
holds for parts (i) and (ii) of Proposition 2, as well as Proposition 8.
To get an analogue of Theorem 1, the cost κ̃ must satisfy additional properties. The most obvious such property is strict convexity: if κ̃ is strictly convex, every rationalizable SCR is uniquely
rationalizable, and so the logic behind Proposition 1 delivers a uniformly dense set of SCRs thatare uniquely rationalizable.
With strict convexity, one can also get an analogue of Theorem 2, provided κ̃ is sufficiently
smooth. Say u ∈ U is a derivative of κ̃ at s if for all t,d+s κ̃(t) = E [u · (t − s)] .
Similar to the case in which C is iteratively differentiable, we show in the appendix that given aninterior s, the cost κ̃ admits u as a derivative at s only if all utilities that rationalize s differ from u
by a nuisance term; that is, v rationalizes s if and only if some λ ∈ L1 (µ0 ) is such that va = ua + λfor all a. Armed with this observation, one can repeat the arguments guaranteeing Theorem 2 toshow a weak*-dense set of SCRs exists that induce unique subset predictions, provided κ̃ is finitevalued, admits a derivative at any interior SCR, and has infinite slope at the edges. Thus, whereasκ̃ imposes few restrictions on the agent’s behavior in a given menu, across menus, one can still useκ̃ to make meaningful predictions about the agent’s choices.

31

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Tech. rep., Mimeo.
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Policy, vol. 49. Elsevier, 317–356.
———. 2003. “Implications of Rational Inattention.” Journal of Monetary Economics 50 (3):665–
690.
———. 2006. “Rational Inattention: Beyond the Linear-Quadratic Case.” American Economic
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Srivastava, Sashi Mohan. 2008. A Course on Borel Sets, vol. 180. Springer Science & BusinessMedia.
Winkler, Gerhard. 1988. “Extreme Points of Moment Sets.” Mathematics of Operations Research
13 (4):581–587.

A. Proof Appendix
A.1. Section 3 Proofs
Proof of Lemma 1. First, suppose p  ps , as witnessed by m : ∆Ω → ∆∆Ω. Any a ∈ supp(s)then has m(·|µsa )  p because p is a proper weighted average of the finitely many measures{m(·|µsa )}a∈supp(s) . So let αa : ∆Ω → [0, 1] be some version of the scaled Radon-Nikodyms)afor each a ∈ A with psa > 0; and let αa = 0 for every other a ∈ A. By
derivative psa dm(·|µdpPconstruction, a∈A αa =p-a.e. 1, so we can change {αa }a∈A on a p-null set to ensure the equationP
holds globally. Let us now show the strategy (p, α) induces s, where α := a∈A αa δa . Indeed, for
any action a ∈ A and event Ω̂ ⊆ Ω, the strategy’s induced probability of action a being played andevent Ω̂ occurring is zero (like under s) if psa is zero, and is otherwise equal toZµ(Ω̂)αa (µ) p(dµ) =Zs)aµ(Ω̂)psa dr(·|µ(µ) p(dµ) = psadpZµ(Ω̂) m(dµ|µsa ) = psa µsa (Ω̂) = E [1Ω̂ sa ] .

Therefore, p can induce s.
Conversely, suppose some strategy (p, α) induces s. For any a ∈ A with psa > 0, we can defineR
q a ∈ ∆∆Ω by letting q a (D) := p1s D α(a|µ) p(dµ) for every Borel D ⊆ ∆Ω. Every a ∈ supp(s)aand every event Ω̂ ⊆ Ω then haveZaµ(Ω̂) q (dµ) = p1saZµ(Ω̂)α(a|µ) p(dµ) = p1saZΩ̂sa (ω) µ0 (dω) = µsa (Ω̂).

Said differently, every a ∈ supp(s) has µ q a (dµ) = µsa . Hence, p =
ps .
R

P

P
s a
s
a∈A pa q a∈A pa δµsa =For the proof that follows, recall the Hardy-Littlewood-Polya-Blackwell-Stein-Sherman-Cartiertheorem (Phelps, 2001, p. 94)—hereafter, the HLPBSSC theorem—says p, q ∈ P satisfy p  q
RR
if and only if f p(dµ) ≥ f q(dµ) for every convex continuous f : ∆Ω → R.
Our next lemma establishes several connections between stochastic choice rules and their revealed information policies. This lemma is the crucial step required for proving Lemma 2.
Lemma 5. The following hold:
36

n

s ∞
s 37
(i) If (sn )∞n=1 weak* converges to s, then (p )n=1 converges to p .

(ii) For any s, t in S and τ ∈ (0, 1),(1 − τ )ps + τ pt  p(1−τ )s+τ t .

(9)(iii) Moreover, the information ranking in (9) is strict whenever some a ∈ supp(s) ∩ supp(t)exists such that µsa , µta .
Proof. Let us first prove part (i). Recall sn → s weak* in S tells us
n
psa

Z

n
f (ω) µsa (dω) → psaZ

f (ω) µsa (dω) as n → ∞, ∀a ∈ A and f ∈ L1 (µ0 ).
nConsequently, any a ∈ A (specializing to f = 1) has psa → psa ; and any a ∈ supp(s)—scaling aRRsnngiven f ∈ L1 (µ0 ) by ppas , which converges to 1—has f (ω) µsa (dω) → f (ω) µsa (dω) for everyaf ∈ L1 (µ0 ). Because every continuous f : Ω → R represents some element of L1 (µ0 ), the latternnproperty tells us µsa → µsa in ∆Ω if a ∈ supp(s). Hence, ps → ps in P, as desired.
Now, we turn to parts (ii) and (iii)] Let r := (1 − τ )s + τ t and p := (1 − τ )ps + τ pt Directcomputation shows supp(r) = supp(s) ∪ supp(t) and, for every a ∈ supp(r), we have
s

t

)pa s
pra = (1 − τ )psa + τ pta and µra = (1−τ
µa + τppra µta .
praa

Observe now that any convex continuous f : ∆Ω → R has
Zf (µ) p(dµ) −Zf (µ) pr (dµ) =

h

X





t
(1−τ )psaf (µsa ) + τppra f
pra
a





t
(1−τ )psa sµa + τppra µta
pra
a



i

(1 − τ )psa f (µsa ) + τ pta f µta − pra f (µra )a∈supp(r)

=

pra

X
a∈supp(r)

≥

h 

pra f

X
a∈supp(r)=hiµta − f (µra )i− f (µra )pra [f (µra ) − f (µra )] = 0.

Xa∈supp(r)The HLPBSSC theorem then implies p  pr , delivering part (ii).
Toward (iii), suppose some a ∈ supp(s) ∩ supp(t) is such that µsa , µta . Now, specialize theabove algebra to the case in which f |co{µsa ,µta } is strictly convex.38 The inequality in the chain isRR
then strict, witnessing f (ω) p(dω) − f (ω) pr (dω) > 0 so that p  pr . The result follows. nA sequence (sn )∞n=1 weak* converges to s if E [ua sa ] converges to E [ua sa ] for all u ∈ U and a ∈ A. Note theweak* topology on S ⊂ S is determined by its convergent sequences because the predual U is separable.
R2
38
g(ω) µ(dω) for some continuous g : Ω → R withR For sinstance, Rf couldt be given by f (µ) :=g(ω) µa (dω) , g(ω) µa (dω).
3737

Now, we prove the indirect cost inherits the information cost’s regularity properties.
Proof of Lemma 2. Because C is proper, any constant SCR has finite cost, and so κ is proper. To
prove that κ is convex and weak* lower semicontinuous, recall κ(s) = C(ps ).
To see lower semicontinuity, consisder any sequence (sn )∞n=1 of SCRs converging to s. BynLemma 5(i), lim inf n→∞ κ(sn ) = lim inf n→∞ C(ps ) ≥ C(ps ) = κ(s), where the inequality
follows from lower semicontinuity of C.
Toward convexity, take any s, t ∈ S and τ ∈ (0, 1). Lemma 5(ii) impliesκ((1 − τ )s + τ t) = C(p(1−τ )s+τ t )≤ C((1 − τ )ps + τ pt )(10)

≤ (1 − τ )C(ps ) + τ C(pt ) = (1 − τ )κ(ps ) + τ κ(pt ),where monotonicity of C implies the first inequality, and convexity of C delivers the second.



A.2. On the Value Function
Extend κ to S by setting κ(s) = ∞ for all s ∈ S \ S. The goal of this section is to prove someresults regarding the optimal value function V : U → R defined as
V (u) := max [E[u · s] − κ(s)] = max [E[u · s] − κ(s)] .
s∈S

s∈S

Our results also pertain to the subdifferential of V at a utility u,∂V : U ⇒ S,
u 7→ {s ∈ S : V (v) ≥ V (u) + E [(v − u)s] for all v ∈ U} .
For any subset T ⊆ S, define the upper and lower inverses of ∂V :∂V U (T ) := {u ∈ U : ∂V (u) ⊆ T },∂V L (T ) := {u ∈ U : ∂V (u) ∩ T , ∅}.
We say ∂V is norm-to-norm (resp. norm-to-weak*) upper hemicontinuous if ∂V U (T ) is norm
open whenever T is norm (resp. weak*) open.
The next lemma establishes continuity properties of the value function and its subdifferential.
Lemma 6. The value function V : U → R is convex and norm continuous, and its subdifferential
is norm-to-weak* upper hemicontinuous.

38

Proof. The value function is real-valued because (as noted in the main text, by Banach-Alaoglu)
each u ∈ U admits some u-rationalizable SCR. It is convex as a supremum of affine functions.
To show V is norm-continuous, we need only show (Aliprantis and Border, 2006, Theorem
5.43) it is bounded above on some ball. And indeed, any u ∈ U with kuk1 ≤ 1 has
V (u) ≤

maxs∈S: κ(s)<∞E[u · s] ≤ kuk1 max ksk∞ = kuk1 ≤ 1.
s∈S

Finally, upper hemicontinuity of ∂V then follows from Proposition 6.1.1 of Borwein and Vanderwerff (2010).

Next, we collect some standard facts from convex analysis, applied to our setting. Given s ∈ S κ
00
and s0 ∈ S, let d+s κ(s ) denote the directional derivative of κ at s in direction s − s.
Lemma 7. Viewing U with its norm topology and S with its weak* topology, V is the convex
conjugate of κ, and κ is the convex conjugate of V . Moreover, the following are equivalent:(i) s ∈ argmaxt∈S [E[u · t] − κ(t)] .
(ii) u ∈ ∂κ(s).
00(iii) Every s0 ∈ S κ has d+s κ(s ) ≥ E [u · (s − s)].

(iv) s ∈ ∂V (u).
Proof. Recall U with its norm topology and S with its weak* topology form a dual pair withthe bilinear map (u, t) 7→ E [u · t]. With these respective topologies, κ is proper, convex, andlower semicontinuous (by Lemma 2). By definition of V , it equals the convex conjugate of theindirect cost function; that is, V = κ∗ . Hence, by the Fenchel-Moreau theorem (e.g., Borwein and
Vanderwerff, 2010, Proposition 4.4.2), κ is the convex conjugate of V ; that is, κ = V ∗ .
Now, we pursue the four-way equivalence. First, that (i) is equivalent to (ii) is immediate (seediscussion after the statement of Lemma 4). Next, Aliprantis and Border’s (2006) Theorem 7.16
directly implies (ii) is equivalent to (iii). Finally, to show (i) is equivalent to (iv), Borwein and
Vanderwerff’s (2010) Proposition 4.4.1 part (a) delivers that s ∈ ∂V (u) holds if and only if
V (u) + κ(s) = E[u · s],
which is equivalent toE[u · s] − κ(s) = V (u) = max [E[u · t] − κ(t)] .
t∈SIt follows (iv) is equivalent to (i), as desired.
39

The following lemma provides sufficient conditions for a set of SCRs to have its rationalizing
utilities be an open set.
Lemma 8. Suppose T ⊆ S is weak* open, and every u ∈ U and u-rationalizable s ∈ T are suchthat s is uniquely u-rationalizable. Then, the set of utilities that rationalize SCRs in T is open.
Proof. Recall u ∈ U rationalizes s ∈ S if and only if s ∈ ∂V (u) (Lemma 7). It follows the set
of utilities that rationalize the SCRs in T is given by ∂V L (T ). Moreover, because ∂V is norm-toweak* upper hemicontinuous (Lemma 6) and T is weak* open, we know ∂V U (T ) is norm open.
Hence, the lemma will follow if we can establish that ∂V L (T ) = ∂V U (T ).
Toward showing ∂V L (T ) = ∂V U (T ), we use the fact that u rationalizes s ∈ T if and only if
s is uniquely u-rationalizable. Therefore, ∂V (u) ∩ {s} , ∅ if and only if ∂V (u) ⊆ {s}, meaning∂V L ({s}) = ∂V U ({s}) must hold for all s ∈ T . As such,∂V L (T ) = ∪s∈T ∂V L ({s}) = ∪s∈T ∂V U ({s}) ⊆ ∂V U (T ) ⊆ ∂V L (T ),where the last containment follows from ∂V being nonempty-valued. So ∂V L (T ) = ∂V U (T ).


A.3. Section 4 Proofs
Proof of Proposition 1. Recall from Lemma 7 that s ∈ S is rationalizable if and only if κ is subdifferentiable at s, that is, if and only if ∂κ(s) , ∅. Also from that lemma, κ is the convex conjugateof the function V defined on the Banach space U. Moreover, V is proper, convex, and continuous (Lemma 6). Thus, the dual Brøndsted-Rockafellar theorem (Brøndsted and Rockafellar, 1965,
second part of Theorem 2) says κ is subdifferentiable on a norm-dense subset of its domain S κ .
The second point holds because (Theorem 23.4 in Rockafellar, 1970) a proper convex function
on a Euclidean space is subdifferentiable everywhere in the relative interior of its domain.
The next lemma shows κ is “almost” strictly convex if additional information is never free.
Lemma 9. If C is strictly monotone, and s, t ∈ S and a ∈ A are such that sa and ta are notproportional, then κ is strictly convex on co{s, t}.
Proof. Recall Lemma 1 tells us κ(r) = C(pr ) for every r ∈ S.
Take any τ ∈ (0, 1), and let r := (1 − τ )s + τ t and p := (1 − τ )ps + τ pt . Part (iii) from
Lemma 5 shows p  pr . Hence,κ(r) = C(pr ) < C(p) ≤ (1 − τ )C(ps ) + τ C(pt ) = (1 − τ )κ(s) + τ κ(t),where the inequalities come from strict monotonicity and convexity of C, respectively.
40Next, we show κ is strictly convex through linearly independent SCRs if C is strictly monotone.
Proposition 9. Suppose C is strictly monotone. If s is linearly independent, then κ is strictly
convex through s.39 Consequently, if s is u-rationalizable, it is uniquely u-rationalizable.
Proposition 9 identifies the set of linearly independent SCRs as ones that cannot be rationalized using indifference when C is strictly monotone. This result generalizes a well-known condition for unique rationalizability under mutual information costs. Specifically, Caplin and Dean(2013) and Matějka and McKay (2015) show that, with mutual information costs, s is uniquelynou-rationalizable whenever the set eua : a ∈ A consists of |A| affinely independent elements. To
see why this result is a specialization of Proposition 9, recall that Matějka and McKay (2015) show
that,40 with mutual information costs, s is optimal when the utility is u only if
psa eua (ω)
s ub (ω).
b∈A pb esa (ω) = Pno

One easily verifies any s satisfying this equation is linearly independent whenever eua : a ∈ A
are |A| affinely independent elements. Unique rationalizability then follows from Proposition 9.
We now turn to proving Proposition 9.

Proof of Proposition 9. The second assertion follows immediately from the first, because the expected benefit from a stochastic choice rule is an affine function of the stochastic choice rule, anda strictly concave function can have at most one maximizer. We turn now to the first assertion.
Given that κ is already known to be weakly convex by Lemma 2, we need only show, given
arbitrary t ∈ S \ {s}, that κ is strictly convex on co{s, t}.
So suppose t ∈ S with κ not strictly convex on co{s, t}. Lemma 9 then tells us ta is a scalarmultiple of sa (which is assumed to be nonzero) for every a ∈ A. Equivalently, µta = µsa for everya ∈ supp(t). Hence,
X
X
X
pta µsa =pta µta = µ0 =psa µsa .
a∈A

a∈A

a∈A

Affine independence of the |A| beliefs {µsa }a∈A then implies pta = psa for every a ∈ A, so thatt = s, delivering the result.



Although not relevant to our subsequent results, we briefly note a stronger uniqueness property(proven in Appendix C) that follows readily for the special case of binary actions.
Corollary 2. Suppose |A| = 2 and C is strictly monotone, and fix u ∈ U. Either a unique SCR is urationalizable, or every u-rationalizable SCR generates state-independent behavior. In particular,
all optimal strategies entail the same information policy.
39
40

That is, κ is strictly convex on any line segment in S κ that includes s.
See also Csiszár (1974).

41

The following lemma shows the affine independence case of Proposition 9 is the typical case.
Lemma 10. If |Ω| ≥ |A|, then supp(ps ) consists of |A| affinely independent beliefs—that is, s islinearly independent—for a weak*-open and uniformly dense set of s ∈ S.
Proof. Define Ŝ to be the set of all s ∈ S such that a∈A sa = 1.
FLet Ω = a∈A Ωa be a Borel partition into sets with positive µ0 -measure; such a partition exists
because Ω is metrizable with at least |A| distinct points and µ0 has full support. Then, define
the set M̂ ⊆ RA×A as the set of matrices for which the a row’s entries sum to µ0 (Ωa ) for eachhRia ∈ A, and the function π : Ŝ → M̂ given by π(s) := Ωa sã (ω) µ0 (dω). The map π isa,ã∈Aaffine, weak* continuous, and hence norm continuous, and, since {Ωa }a∈A are pairwise disjoint
with nonzero measure, surjective. Because both Ŝ and M̂ are closed affine subspaces of Banach
spaces, it follows from the open mapping theorem that π is a norm-open map: it maps norm-open
sets to open sets. Hence, if G is any open subset of M̂ with closure equal to π(S), then π −1 (G) is
weak* open in Ŝ (because π is weak* continuous) with norm closure containing S (because π isnorm open).41 It therefore suffices to find some open G ⊆ M̂ with closure equal to π(S) such thatany given s ∈ S ∩ π −1 (G) has supp(ps ) consisting of |A| affinely independent beliefs.
To that end, let G be the convex set of invertible matrices in M̂ with strictly positive entries.
The set G is open in M̂ because the determinant function is continuous. Moreover, it is contained in
π(S), which is the set of matrices in M̂ with nonnegative entries. Next, to see G is nonempty, defineihg := [ + (µ0 (Ωa ) − |A|) 1a=ã ]a,ã∈A ∈ M̂ for  ∈ 0, mina∈A|A|µ0 (Ωa ) . Because g0 is diagonal with
nonzero diagonal entries, its determinant is nonzero. Thus, since a nonzero univariate polynomial
has only finitely many roots, det(g ) is nonzero for  > 0 sufficiently small. The matrix g :=g belongs to G for such an . Now, we observe G is dense in π(S). Fixing an arbitrary h ∈π(S), we want to show h is a limit of matrices from G. Define the function R → R via γ 7→
det [γg + (1 − γ)h] and note it is a polynomial that is not globally zero (by evaluating at γ = 1).
Hence, the polynomial has only finitely many roots. In particular, some γ̄ ∈ (0, 1) exists such thatγg + (1 − γ)h ∈ M̂ is invertible for every γ ∈ (0, γ̄)—and so h is in the closure of M̂ .
All that remains is to show, for any given s ∈ S ∩ π −1 (G), that supp(ps ) consists of |A|affinely independent beliefs. Toward showing this property, observe psa > 0 for every a ∈ Abecause π(s) has nonnegative entries and no zero columns. We can now define the matrix m :=
h R
i
1
s dµ0
= [µsã (Ωa )]a,ã∈A . Because m is a product of π(s) and another invertible matrix,psã Ωa ã
a,ã∈Aits columns are linearly independent, and therefore affinely independent. It follows directly thatthe |A| vectors {µsa }a∈A are affinely independent, as required.
P

41

To see the latter implication, let T be any norm-open set in Ŝ that intersects S. Observe that because T intersects
S, the set π(T ) intersects π(S) too. Note also that π(T ) is open in M̂ , because π is norm open. Therefore, π(S) is
contained in the norm closure of G, and so G intersects π(T ). Said differently, T intersects π −1 (G), as required.

42

We next show Assumption A1(ii) is needed for unique rationalizability with locally affine costs.
Claim 1. Suppose s ∈ S κ is such that |supp s| > |Ω|. If C is affine in a neighborhood of ps , s is
not uniquely rationalizable.
Claim 1 and Proposition 9 are related to Sharma, Tsakas, and Voorneveld (2020). Their Proposition 1(b) shows that with binary states and strictly monotone posterior separable costs, a rationalizable information policy is uniquely rationalizable if and only if it has at most binary support.
Our Proposition 9 generalizes their “if” direction, since a set of beliefs is affinely independent if
and only if its cardinality is at most 2. Claim 1 generalizes the “only if” direction, because strict
monotonicity of C implies every rationalizable p must induce an SCR with a support of weakly
larger size than |supp p|.
Proof of Claim 1. Take u ∈ U and s ∈ S. Suppose s is the unique stochastic choice rule induced
by some optimal strategy given u. We must show |supp(s)| ≤ |Ω|. The result is vacuous (given|A| < ∞) when Ω is infinite, so we focus on the case in which Ω is finite.
First, let us establish sa and sã cannot be proportional for any two distinct a, ã ∈ supp(s).
For a contradiction, assume they are in fact proportional. For any τ ∈ [0, 1], then, we can definesτ ∈ S via sτa := (1 − τ )(sa + sã ), sτã := τ (sa + sã ), and sτa0 := sa0 for every a0 ∈ A \ {a, ã}. Byτ
construction, ps is the same for every τ ∈ [0, 1], so that Lemma 1 implies κ(sτ ) is the same forevery τ ∈ [0, 1]. Therefore, the objective τ 7→ E [u · sτ ] − κ(sτ ) is affine because τ 7→ sτ is. It
psã
follows this objective cannot be uniquely maximized at τ s = ps +ps ∈ (0, 1)—contradicting uniqueaã
optimality of s because different values of τ generate different stochastic choice rules.
Having shown sa and sã cannot be proportional for any two distinct a, ã ∈ supp(s), we know{µsa }a∈supp(s) are |supp(s)| distinct elements of ∆Ω. The set ∆Ω can have no more than |Ω| affinely
independent vectors, so the claim will follow if we show {µsa }a∈supp(s) are affinely independent.
Assume, for a contradiction, supp(ps ) = {µsa }a∈supp(s) are affinely dependent. By Winkler’s(1988) Theorem 2.1(b), we then have ps < ext(P). So ps is the midpoint of two distinct information
policies p1 , p2 ∈ P. Replacing each of p1 , p2 with a weighted average with ps if necessary, we may
assume C is affine on co{p1 , p2 }. Optimality of s implies the agent’s optimal value is


E


X

ua sa  − C(ps ) =

X
i=1,2

a∈supp(s)



1

2


X

pi (µsa )

ua dµsa  − C(pi ) .
a∈supp(s)

hP

Z

i

i
s
s
i
Hence, some i ∈ {1, 2} has v :=a∈supp(s) p (µa ) ua dµa − C(p ) weakly higher than theagent’s optimal value. Payoff v is clearly attainable via some strategy that induces stochastic choicei srule si ∈ S given by sia := p p(µs a ) sa for a ∈ supp(s) and sia := 0 for a ∈ A \ supp(s). Hence, si isarationalizable, in contradiction to s being uniquely so. The claim follows.


R

43

Proof of Theorem 1. Let R ⊆ S be the set of all rationalizable SCRs if Ω is infinite, and let it be
the relative interior of S κ (which is the interior of S κ in S by Assumption A1(iii)) if Ω is finite. By
Proposition 1, R is norm dense (i.e., uniformly dense) in S κ , and every SCR in R is rationalizable.
So, R is a norm-dense subset of S κ with only rationalizable SCRs, and R is open in S if Ω is finite.
Now, Lemma 10 delivers a weak*-open (and hence norm-open) and norm-dense subset G of
S such that every s ∈ G has supp(ps ) consisting of |A| affinely independent beliefs. By Proposition 9, every u ∈ U and u-rationalizable s ∈ R ∩ G are such that s is uniquely u-rationalizable.
Moreover, R ∩ G is a norm-dense subset (being the intersection of two norm-dense subsets, one ofwhich is open) of S κ that is also open in S κ whenever R is (which is true whenever Ω is finite).
It remains to show the set of utilities that rationalize the SCRs in R ∩ G is open. But theseutilities are the ones that rationalize SCRs in G (resp. R ∩ G) if Ω is infinite (resp. finite)—aweak*-open set in either case—and so Lemma 8 applies directly.

Proof of Proposition 2. Here, we prove the following three claims:(i) Every rationalizable SCR is continuous if and only if every SCR in S κ is continuous.
(ii) If the domain S κ has a nonempty (norm) interior in S, then every rationalizable SCR iscontinuous if and only if |Ω| < ∞.
(iii) With Assumption A1, each uniquely rationalizable SCR is continuous if and only if |Ω| < ∞.
Observe (i) corresponds to the statement of Proposition 2. We now proceed with the proof.
κFirst, observe the set Scontof stochastic choice rules in S κ with a continuous version is k · k∞ closed in S κ . To see it is closed, note that because µ0 has full support, the natural quotient taking acontinuous function to its µ0 -a.e. equivalence class is an isometry from the space of of continuousfunctions Ω → R (with the supremum norm) into L∞ (µ0 ). Because the space of continuousfunctions is complete, its image under this isometry is necessarily closed in L∞ (µ0 ).42 Hence,κ
being the intersection of S κ with a power of this set, the set Scontis closed in S κ .
κκNow, because Scontis closed, any given dense subset of S κ is contained in Scont
if and onlyκκ
if Scont = S . In particular, Proposition 1 (resp. Theorem 1 under Assumption A1) implies
κκ
every (uniquely) rationalizable stochastic choice rule lives in Scont
if and only if Scont= S κ . Thisobservation delivers item (i) (hence Proposition 2 as stated). It would also deliver items (ii) and (iii)
if we could show that, assuming S κ has a nonempty interior, every element of S κ has a continuousversion if and only if Ω is finite.
42

To explain, take any sequence (fn )n of continuous functions whose equivalence classes converge to some g (andso form a Cauchy sequence). The sequence (fn )n of functions themselves form a Cauchy sequence by isometry. But
then, by completeness, (fn )n converges to some continuous function f . Again by isometry, the equivalence class of
fn converges to that of f as n → ∞. Because the metric space L∞ (µ0 ) is Hausdorff, g is in fact f ’s equivalence class,and so lives in the image of this isometry.

44Suppose S κ has nonempty interior. Our goal is now to show every element of S κ has a continuous version if and only if Ω is finite. One direction of this equivalence is trivial, because every
map on a finite metric space is continuous. Toward establishing the converse, suppose Ω is infinite.
κwould exist if we had some Borel f : Ω → [0, 1] that is not
Let us now observe some s ∈ S κ \ Scontµ0 -a.e. equal to a continuous function. Indeed, because S κ has a nonempty interior in S, we knowsome s0 ∈ S κ and  > 0 exist such that S κ ⊇ {s ∈ S : |sa − s0a | ≤µ0 -a.e. , ∀a ∈ A}. Shifting s0
and reducing  if necessary, we may further assume  ≤µ0 -a.e. sa ≤µ0 -a.e. 1 −  for every a ∈ A.
Then, fix two distinct a+ , a− ∈ A, and define s1 ∈ S by letting s1a+ := s0a+ + f , s1a− := s0a− − f ,and s1a := s0a for every a ∈ A \ {a+ , a− }. By construction, s1 ∈ S κ too, and s1 − s0 has nocontinuous version. Therefore, at least one of {s0 , s1 } has no continuous version, as desired.
All that remains now is to construct (for infinite Ω) some Borel f : Ω → [0, 1] that is not µ0 a.e. equal to a continuous function. Because Ω is an infinite compact metrizable space, it has somesequence {ωn }∞n=1 without repetition that converges to some ω∞ . In particular, ω∞ is the soleaccumulation point of {ωm }m∈N∪{∞} . Dropping to a subsequence if necessary, we may assume
ωn , ω∞ for every n ∈ N. Now, take some ρ that metrizes the topology on Ω. For each n ∈ N, let
rn > 0 be such that the ρ(ωn , ωm ) > 2rn for every m ∈ (N ∪ {∞}) \ {n}, and let Nn be the ρ-ballof radius rn centered on ωn . By the triangle inequality, the neighborhoods {Nn }∞n=1 are pairwise
disjoint. Moreover, that ωn → ω∞ implies rn → 0 as n → ∞. Define now the bounded Borel
function f := 1S∞ N2n : Ω → [0, 1], which we show has no continuous version. To do so, taken=1
an arbitrary Borel function f˜ : Ω → R that is µ0 -a.e. equal to f . To conclude the proof, we showf˜ cannot be continuous. For any n ∈ N, that µ0 has full support implies µ0 (Nn ) > 0, and so someω̃n ∈ Nn has f˜(ω̃n ) = f (ω̃n ). Then, ρ(ω̃n , ω∞ ) ≤ rn + ρ(ωn , ω∞ ) → 0 as n → ∞. However,f˜(ω̃n ) = f (ω̃n ) is equal to 0 for odd n and 1 for even n, so that f˜ is discontinuous at ω∞ .


A.4. On Differentiable Costs
We begin this subsection with some convenient notations. First, given u ∈ U, letvu : ∆Ω → Rµ 7→ max

Z

a∈A

Ω

ua (ω) µ(dω).

Notation 1. Let feas C be the (convex) set of p ∈ P F ∩ P C such that every q ∈ P F admits some ∈ (0, 1) with p + (q − p) ∈ P C .
Analogously, for any c ∈ C, let feas c be the (convex) set of simply drawn posteriors µ suchthat every simply drawn posterior µ̃ admits some  ∈ (0, 1) with c (µ + (µ̃ − µ)) < ∞.

45

Note feas C = P F if C assigns a finite cost to every simple information policy; and feas c is theset of all simply drawn posteriors whenever c assigns a finite cost to every simply drawn posterior.
The following lemma gives an equivalent optimality condition for an SCR: the agent responds
optimally to any revealed posterior, and, assuming the agent responds optimally to any hypotheticalposterior, the information is optimal.
Lemma 11. Given u ∈ U and s ∈ S, the following are equivalent:(i) The stochastic choice rule s is optimal; that is, s ∈ argmaxt∈S κ {E [u · t] − κ(t)}.
(ii) Every a ∈ supp(s) has ua (ω) µsa (dω) = vu (µsa ), andRsp ∈ argmaxp∈P C

Zvu (µ) p(dµ) − C(p) .

Proof. Let αu : ∆Ω → A be a measurable selection of µ 7→ argmaxa∈A ua (ω) µ(dω), which
exists by the measurable maximum theorem (Aliprantis and Border, 2006, Theorem 18.19).
R
To see (i) implies (ii), we suppose (ii) does not hold. Because vu (µ) ≥ ua (ω) µ(dω) forR
all a ∈ A and µ ∈ ∆Ω, if vu (µsa ) , ua (ω) µsa (dω) for some a ∈ supp(s), it must be thatRvu (µsa ) > ua (ω) µsa (dω), and so the SCR t induced by (ps , αu ) is a strict improvement over s.
R
R
Similarly, if ps < argmaxp∈P C [ vu (µ) p(dµ) − C(p)], one can pick p ∈ P C with vu (µ) p(dµ) −R
C(p) > vu (µ) ps (dµ) − C(ps ), in which case the SCR t induced by (p, αu ) is strictly better thans. Either way, (i) fails.
To see (ii) implies (i), suppose (ii) holds. Then, any SCR t hasR

E [u · t] − κ(t) = E [u · t] − C(pt ) (by Lemma 1)




= 

pta

X

Z

ua (ω) µta (dω) − C(pt )a∈supp(pt )





≤ 

pta vu (µta ) − C(pt )Xa∈supp(pt )=Zvu (µ) pt (dµ) − C(pt ),where the inequality holds with equality for t = s by hypothesis. Then, s maximizes t 7→ E [u · t]−Rκ(t) because ps maximizes p 7→ vu (µ) p(dµ) − C(p).
Our definition of a derivative of C assumes the derivative is convex, making the posteriorseparable approximation Cc a valid cost function. Here, we note that under sufficient regularity,
this convexity property is redundant because C is monotone. See Appendix C for a proof.
46

Fact 1. Let p ∈ feas C, and let c : ∆Ω → R be a continuous function. If every p0 ∈ P C hasZ0d+p C(p ) =c(µ) (p0 − p)(dµ),

then c is convex. In particular, in this case, c ∈ C, so that c is a derivative of C at p.
The following lemma establishes an equivalence between optimal information choice for theagent’s cost function and for a posterior-separable approximation of her cost function.
Lemma 12. If c is a derivative of C at p ∈ feas C, then any u ∈ U hasZp ∈ argmaxq∈Pvu (µ) q(dµ) − C(q)⇐⇒ p ∈ argmaxq∈PZ[vu (µ) − c(µ)] q(dµ).

Proof. Suppose first p ∈ argmaxq∈P [vu (µ) − c(µ)] q(dµ). Then, for every q ∈ P C ,RZvu (µ) (q − p)(dµ) ≤Zc(µ) (q − p)(dµ) ≤ C(q) − C(p),where the last inequality follows from C (p + (q − p)) being convex in  ∈ [0, 1]. Because C(q) =∞ for all q ∈ P \ P C , the left-hand-side condition follows.
R
R
Conversely, suppose some p̃ ∈ P has [vu (µ) − c(µ)] p̃(dµ) > [vu (µ) − c(µ)] p(dµ). Because c is convex and lower semicontinuous, and vu is the maximum of finitely many affine functions, we may assume without loss that p̃ ∈ P F . Observe every  ∈ (0, 1) satisfiesZ[vu (µ) − c(µ)] [p + (p̃ − p)] (dµ) >Z[vu (µ) − c(µ)] p(dµ).

Thus, because p ∈ feas C, we may assume without loss some convex combination p0 of p̃ and p
0
has p ∈ P C . But then
0<

Z

0

vu (µ) (p − p)(dµ) −

1
= lim
&0 

Z

Z

c(µ) (p0 − p)(dµ)

00vu (µ) [p + (p − p)] (dµ) − C (p + (p − p)) −Zvu (µ) p(dµ) + C(p) ,so that small enough  ∈ (0, 1) will satisfy
Z

vu (µ) [p + (p0 − p)] (dµ) − C (p + (p0 − p)) >Zvu (µ) p(dµ) − C(p).
The lemma follows.

The next lemma establishes an equivalence between optimality of an SCR for our agent’s costfunction and its optimality for a posterior-separable approximation of the same.
47

Lemma 13. Suppose s ∈ S has ps ∈ feas C and c is a derivative of C at ps . Given u ∈ U,s ∈ argmaxs0 ∈S [E [u · s0 ] − κ(s0 )] ⇐⇒ s ∈ argmaxs0 ∈S [E [u · s0 ] − κc (s0 )] .
Proof. Lemma 11 says s is optimal with cost κ if and only if ua (ω) µsa (dω) = vu (µsa ) for all a ∈R
supp(s) and ps ∈ argmaxp∈P C [ vu (ω) p(dω) − C(p)] . Further, the information cost Cc satisfies
our standing hypotheses on C. Lemma 11 therefore tells us s is optimal with cost κc if and onlyRR
if ua (ω) µsa (dω) = vu (µsa ) for all a ∈ supp(s) and ps ∈ argmaxq∈P C [ vu (µ) q(dµ) − Cc (q)] .
R
The equivalence would therefore follow if we knew ps ∈ argmaxp∈P C [ vu (µ) p(dµ) − C(p)] if
and only ifZsvu (µ) p(dµ) − Cc (p) ,p ∈ argmaxp∈P C
Rwhich Lemma 12 guarantees.

The next lemma gives an explicit formula for the directional derivatives of the indirect costfunction when information costs are posterior separable.
Lemma 14. If c ∈ C and s, s0 ∈ S are such that every supp(ps ) ⊆ feas c, then430d+s κc (s ) =

0

h

0

0

i

s
(psa − psa ) c(µsa ) + psa d+µsa c(µa )Xa∈supp(s)00psa c(µsa ).

X+a∈supp(s0 )\supp(s)Proof. Let Â := supp(s) ∪ supp(s0 ), and define s := s + (s0 − s) for each  ∈ (0, 1). Any ∈ (0, 1) has supp(s ) = Â and, for each a ∈ Â,0

= psa + (psa − psa )= µsa +  ppsa (µsa − µsa ).

psa
µsa

s0

0

a

Therefore, every  ∈ (0, 1) and a ∈ Â have
h



0

i



h





i

ηa () := 1 psa c(µsa ) − psa c(µsa ) = (psa − psa ) c(µsa ) + psa 1 c(µsa ) − c(µsa ) .
000Hence, ηa () is equal to psa c(µsa ) if psa = 0, equal to −psa c(µsa ) if psa = 0, and otherwise equal to00(psa − psa ) c(µsa ) + psai
psa 1 h s
s
c(µ
)
−
c(µ
)
0
a
a ,
psa  psa
psa

43

0

0

0

s
When psa = 0, we adopt the convention that psa d+µsa c(µa ) is zero too.

48

which converges as  → 0 (because c is convex, and hence continuous on any open line segment
on its domain, and supp(ps ) ⊆ feas c) to0(psa − psa ) c=lim µsa
&0



psa

0

+ psa

s

lim&0 pa
0
s0
s
s
s0
+
(pa − pa ) c(µa ) + pa 1 dµsa c(µsa ).

i
1 h s+˜(s0 −s)c(µ) − c(µsa )˜&0 ˜

limHence,
lim 1 [κc (s + (s0 − s)) − κc (s)]&0=

lim 1
&0

Z

&0

h

X

=



cd(ps − ps ) = limXηa ()

a∈Â
0

0

0

i

s
(psa − psa ) c(µsa ) + psa d+µsa c(µa )a∈supp(s)

X

+

0

0

psa c(µsa ),a∈supp(s0 )\supp(s)

as desired.
The following lemma shows derivatives preserve finite cost for an information policy.
Lemma 15. If c is a derivative of C at p ∈ P C , then c(µ) p0 (dµ) < ∞ for every p0 ∈ P C .
R

0Proof. Consider any p0 ∈ P C . Convexity of C implies d+p C(p ) ∈ R ∪ {−∞}. Hence, applyingR
the definition of a derivative (including that c(µ) p(dµ) < ∞),Z0c(µ) p (dµ) =Z0c(µ) p(dµ) + d+
p C(p ) < ∞,as required.

A.5. On Iteratively Differentiable Costs
Although not relevant to our subsequent results, let us briefly note a uniqueness property (provenin Appendix C) for iterated differentiability that justifies the notation ∇cµ .
Fact 2. Any function in C admits at most one derivative at a simply drawn posterior.
The following lemma yields a more explicit form (relative to Lemma 14) for directional derivatives of a posterior-separable approximations, given iterative differentiability.
To state a slightly more general version of the result (which will be helpful for one result
described in section 7), we invest in another definition. For any c ∈ C and any simply drawn
posterior µ, define the subdifferential10∂c(µ) := f ∈ L (µ0 ) : c(µ ) ≥ c(µ) +

Z

0

0



f (ω) (µ − µ)(dω) ∀ simply drawn µ ∈ ∆Ω
49

of c at µ. Clearly, if c is differentiable at µ, then ∇cµ ∈ ∂c(µ) because c is convex.
Remark 1. Although we adopt the notation ∂c for parsimony, the above definition is best understood as the subdifferential of a function c̃ : L∞ (µ0 ) → R, where each element of L∞ (µ0 ) isinterpreted as a Radon-Nikodym derivative with respect to µ0 .
Lemma 16. Let c ∈ C and s, s0 ∈ S have supp(ps ) ⊆ feas c, and let fa ∈ ∂c(µsa ) haveR00fa (ω) µsa (dω) for each a ∈ supp(s). Let us,s ∈ U have us,s= fa for a ∈ supp(s) and
a
s,s0
s0
0
ua = c(µa )1 for a ∈ supp(s ) \ supp(s). Then,00d+s κc (s ) ≥ E [(s − s) · u] ,with equality holding if fa = ∇cµsa for every a ∈ supp(s).
0Proof. Let u := us,s . Then, every a ∈ supp(s) has0s0s0
psa d+µsa c(µa ) ≥ pa
0
= psa

=
=

Z
Z

Z

0

fa (ω) (µsa − µsa )(dω)

Z " 0

#

sa (ω) sa (ω)−
fa (ω) µ0 (dω)
psa0
psa

[s0a (ω) − sa (ω)] fa (ω) µ0 (dω) +[s0a (ω) − sa (ω)] u(ω) µ0 (dω) −

Z 



0

psa − psa0
psa − psa

 s (ω)apsa

Zfa (ω) µ0 (dω)

fa (ω) µsa (dω)

0

= E [(s0a − sa )ua ] − (psa − psa ) c(µsa ).
Moreover, the above inequality holds with equality if fa = ∇cµsa for every a ∈ supp(s). Therefore,
Lemma 14 implies0d+s κc (s ) ≥X

E [(s0a − sa )ua ] +

=

0

0

psa c(µsa )a∈supp(s0 )\supp(s)a∈supp(s)XX

E [(s0a − sa )ua ] +

X

h

0

E (s0a − sa ) c(µsa )ia∈supp(s0 )\supp(s)a∈supp(s)0= E [(s − s) · u] ,again with equality if fa = ∇cµsa for every a ∈ supp(s). The result follows.



The next lemma relates the set feas C to the corresponding set of beliefs for its derivatives.
Lemma 17. If c is a derivative of C at p ∈ feas C, then supp(p) ⊆ feas c.
Proof. Consider µ ∈ supp(p), and any simply drawn posterior µ0 ; we must show some proper
convex combination of µ and µ0 belongs to c−1 (R).
50

By hypothesis, p is simple, µ ∈ supp(p), and µ0 is simply drawn. Hence, some finite-support
q, q 0 ∈ ∆∆Ω and τ, τ 0 ∈ (0, 1] exist with p = (1 − τ )q + τ δµ and p0 := (1 − τ 0 )q 0 + τ 0 δµ0 ∈ P C .
Now, that p ∈ feas C implies some  ∈ (0, 1) has (1 − )p + p0 ∈ P C . Define0(1−)τ
τµ + (1−)τ
µ0 ,µ := (1−)τ
+τ 0
+τ 0

and
p := (1 − )(1 − τ )q + (1 − τ 0 )q 0 + [(1 − )τ + τ 0 ] δµ .
Because C is monotone and p  (1 − )p + p0 by construction, p ∈ P C . Hence, Lemma 15
Rtells us c(µ) p (dµ) < ∞, and so too c(µ ) < ∞. The lemma follows.

The following lemma gives an exact optimality condition for a given SCR when information
costs are iteratively differentiable.
Lemma 18. Let s ∈ S have ps ∈ feas C, suppose C is iteratively differentiable at ps with derivative c. For u ∈ U, the following are equivalent:(i) SCR s is u-rationalizable.
h

i

0

0

0

(ii) Every s0 ∈ S has E (us,s − u) · (s0 − s) ≥ 0, where us,s ∈ U has us,s
= ∇cµsa for
a
s,s0
s0
0
a ∈ supp(s) and ua = c(µa )1 for a ∈ supp(s ) \ supp(s)Proof. By Lemma 13, u rationalizes s if and only if u would rationalize s given alternative information cost Cc . Hence, Lemma 7 (applied to the model with cost Cc ) tells us u rationalizes s if and00s)]. Because Lemma 17 says supp(ps ) ⊆ feas c,only if every s0 ∈ S κ has d+s κc (sh ) ≥ E [u · (s −i0s,s0
· (s0 − s) for every s0 ∈ S. The equivalence follows.

Lemma 16 shows d+s κc (s ) = E u
The following lemma characterizes the utilities that can rationalize a given full-support SCR
when information costs are iteratively differentiable. Letting c denote a derivative of C at the SCR,
all such utilities are given by the derivative of c at the corresponding revealed posterior, augmented
by a nuisance term, maybe with an additional payoff penalty for actions not chosen in a given state.
Lemma 19. Let s ∈ S have ps ∈ feas C and supp(s) = A. Suppose C is iteratively differentiable
at ps with derivative c. The following are equivalent for u ∈ U:(i) SCR s is u-rationalizable.
A(ii) Some λ ∈ L1 (µ0 ) and γ ∈ L1 (µ0 )+ exist such that every a ∈ A has
ua = λ − γa + ∇cµsa ,sa γa = 0.
51

Proof. Let us := u−(∇cµsa )a∈A ∈ U. By Lemma 18, u rationalizes s if and only if every s0 ∈ S has
E [−us · (s0 − s)] ≥ 0, or equivalently, s ∈ argmaxs0 ∈S E [us · s0 ]. Hence, u rationalizes s if andonly if µ0 -almost every ω has {a ∈ A : sa (ω) > 0} ⊆ argmaxa∈A usa (ω). But the latter conditionA
holds if and only if some λ ∈ L1 (µ0 ) and γ ∈ L1 (µ0 )+ have us = (λ − γa )a∈A and (γa sa )a∈A = 0:
The “if” direction is immediate, and the “only if” is witnessed by λ := maxa∈A usa .


A.6. Section 5 Proofs
Proof of Lemma 3. Because C is finite on P F , every SCR s has ps ∈ feas C. The theorem therefore follows directly from Lemma 13.
Proof of Proposition 3. Because C is finite on P F , every SCR s has ps ∈ feas C. The proposition
therefore follows directly from Lemma 19.

Let us briefly note the infinite-slope condition, Assumption A2(iii)—which says costs decrease
infinitely steeply as one moves from a not-fully-mixed information policy toward providing no
information—could be equivalently replaced with a more permissive unbounded steepness condition. See Appendix C for the proof.
Fact 3. For any p ∈ P C , the following are equivalent:(i) d+p C(δµ0 ) = −∞.
(ii) inf q∈P C , ∈(0,1) 1 [C(p + (q − p)) − C(p)] = −∞.
The following corollary follows immediately from Lemma 19 (and the weaker version requiring C to be finite on P F follows immediately from Proposition 3).
Corollary 3. Let s ∈ S have ps ∈ feas C and s have conditionally full support. If C is iteratively
differentiable at ps , and u and u0 both rationalize s, some λ ∈ L1 (µ0 ) exists such thatua = u0a + λ ∀a ∈ A.
Next, we show Assumption A2(iii) means an optimal action recommendation from s neverrules out any state.
Lemma 20. Suppose C satisfies Assumption A2(iii). If s is rationalizable, then any a ∈ supp s
has sa strictly positive µ0 -almost surely.
Proof. Take u ∈ U, and suppose s ∈ S admits some a ∈ supp s such that sa is not µ0 -almost
surely strictly positive. We wish to show s is not u-rationalizable.
52

We have nothing to show if s < S κ , so we focus on the case in which s ∈ S κ . Pick somea0 ∈ A and let t be the unique SCR with ta0 = 1. Clearly, pt = δµ0 , and so t ∈ S κ and pt ∈ P C .
For every  ∈ (0, 1), let t = s + (t − s). Then, because ps is not fully mixed,11[κ(t ) − κ(s)] = [C(pt ) − C(ps )]1≤ [C((1 − )ps + pt ) − C(ps )]
1
&0= [C((1 − )ps + δµ0 ) − C(ps )] −−→ −∞,
where the inequality comes from monotonicity of C and Lemma 5(ii), and the limit calculation
comes from Assumption A2(iii). Hence, every u ∈ U has
11
&0{E [u · t ] − κ(t ) − [E [u · s] − κ(s)]} = E [u · (t − s)] − [κ(t ) − κ(s)] −−→ −∞.
Thus, for all sufficiently small  ∈ (0, 1), the agent’s objective must be strictly higher under t thanunder s. That is, s is not rationalizable.
Now, we prove most utilities give rise a unique SCR.
Proof of Lemma 4. Let V denote the set of u ∈ U at which V is Gâteaux differentiable, and recall
V is continuous by Lemma 6. For any u ∈ V, the function V has a unique subgradient at u
(Borwein and Vanderwerff, 2010, Corollary 4.2.5), and so a unique SCR is u-rationalizable.
It therefore remains to show U \ V is meager and shy. That this set is meager follows from
Mazur’s theorem (see Borwein and Vanderwerff, 2010, Theorem 4.6.3), which tells us V is dense
and Gδ , and hence co-meager. To see U \ V is shy, it suffices to show it is Haar null.44
To show U \ V is Haar null, note Theorem 5.44 from Aliprantis and Border (2006) tells us V is
locally Lipschitz. Hence, being second countable, U can be covered by countably many open balls{Un }∞
n=1 with V |Un Lipschitz for each n ∈ N. Moreover, for each n ∈ N, Theorem 4.6.5 from
Borwein and Vanderwerff (2010) tells us Un \ V is Aronszajn null, and hence (by Benyamini andS
Lindenstrauss, 1998, Proposition 6.25) Haar null. Therefore, U \ V = ∞n=1 (Un \ V) is Haar null(by Borwein and Vanderwerff, 2010, Proposition 4.6.1(e)).
The following lemma shows a norm-dense set of uniquely rationalizable SCRs can be leveraged
to find a weak*-dense set of SCRs generating unique subset predictions.
44

Given Fact 2 from Hunt, Sauer, and Yorke (1992) (resp. Proposition 4.6.1(c) from Borwein and Vanderwerff,
2010), U \ V is shy (resp. Haar null) if and only if some compactly supported finite Borel (resp. Radon) measureassigns zero measure to every translation of it. Because every finite Borel measure on the Polish space U is a Radon
measure, U \ V is shy if and only if it is Haar null.

53

Lemma 21. Suppose Assumption A2 holds and some norm-dense S1 ⊆ S κ comprises only uniquely
rationalizable SCRs. Then, the set of SCRs yielding unique subset predictions is weak* dense in S.
Proof. By Lemma 4, every B ∈ A admits a co-meager set of utilities UB ⊆ U that generate aunique prediction over B; that is, each u ∈ UB admits a unique s that is u-rationalizable over B.
It follows the set UA = ∩B∈A UB is a co-meager (and therefore dense) subset of U such that, forevery B, every u ∈ UA uniquely rationalizes some s ∈ S B over B.
Observe now, the set S1 is norm-dense in S because Assumption A2(i) implies S κ = S. Let
S0 denote the set of full-support SCRs, which is weak* open (hence norm open) and norm dense
in S. Therefore, S̃ := S0 ∩ S1 is norm dense in S too. Because every norm-dense set is weak*dense, it suffices to show S̃ is contained in the weak* closure of the SCRs generating unique subset
predictions. To that end, take any s ∈ S̃, and let N ⊆ S be a weak* neighborhood of s; we want
to show some SCR in N generates unique subset predictions.
Let u be a utility that uniquely rationalizes s (over A). Some sequence (un )n∈N from UA
converges to u because UA is dense in U. Let sn denote the SCR rationalized by un for eachn ∈ N. Because sn ∈ ∂V (un ) by Lemma 7, and ∂V norm-to-weak* upper hemicontinuousw∗(Lemma 6) and single valued at u, it follows that sn −→ s as n → ∞. In particular, sn ∈ N for alln sufficiently large.
Next, observe sn has full support (i.e., belongs to S0 ) for large enough n, because S0 is weak*
open in S, and the weak*-limit s has full support. But Lemma 20 tells us any rationalizable
full-support SCR has conditionally full support in light of Assumption A2(iii). Therefore, sn has
conditionally full support for sufficiently large n.
Now, fix n ∈ N large enough that sn ∈ N and sn has conditionally full support; the theorem
will follow if we can show sn generates unique subset predictions. To that end, consider anynû that rationalizes sn . Because sn has conditionally full support, ps is fully mixed, and so C
nis iteratively differentiable at ps by Assumption A2(ii). Moreover, Assumption A2(i) impliesnfeas C ⊇ P F 3 ps , and so Corollary 3 delivers some λ ∈ L1 (µ0 ) for which every a ∈ A hasûa = una + λ. It follows that every t ∈ S κ has
E [û · t] − κ(t) = E [un · t] − κ(t) + E [λ] ,and so every B ∈ A has
argmaxt∈SB [E [û · t] − κ(t)] = argmaxt∈SB [E [un · t] − κ(t)] .
Hence, over any B, the utility û rationalizes the same set of SCRs as un does. But un ∈ UA ,meaning it rationalizes a unique SCR over B. It follows sn yields unique subset predictions, as54

required.

Proof of Theorem 2. Theorem 1 delivers a norm-dense subset of S κ comprising only uniquely rationalizable SCRs. The result then follows directly from Lemma 21.


A.7. Section 6 ProofsDfor every {a, B} ∈ E. For any ` ∈ RE ,Proof of Corollary 1. Let E := ED , and let fa,B := fa,BP
D
let ` · f := {a,B}∈E `{a,B} fa,B
. Observe now that condition (ii) says `χ · f = 0 for every testable
cycle χ, whereas condition (iii) says `χ ·f = 0 for every testable cycle χ in the cycle basis. Because` 7→ ` · f is linear, the equivalence of (ii) and (iii) follows directly.
Now, consider any B ∈ B and any u ∈ U. Because β has conditionally full support, Proposition 3 (applied to the model with action set B) tells us β B is u-rationalizable over B if and only ifsome λB ∈ L1 (µ0 ) exists such that every a ∈ B has ua = λB + fa,B . Condition (i) is thereforeA∪B
equivalent to the existence of (u, λ) ∈ L1 (µ0 )such that

fa,B = ua − λB for every {a, B} ∈ E.

(11)
A∪B

It remains to see (ii) is equivalent to the existence of (u, λ) ∈ L1 (µ0 )satisfying (11).45
A∪B
First, if (u, λ) ∈ L1 (µ0 )satisfies (11) and χ = a0 B1 a1 . . . BN aN is a testable cycle, then
N 
X
n=1



fan ,Bn − fan−1 ,Bn =

N h
X

i

(uan − λBn ) − (uan−1 − λBn ) = uaN − u0 = 0,

n=1

verifying condition (ii). Conversely, suppose (ii) holds.
A
B
Toward constructing u ∈ L1 (µ0 ) and λ ∈ L1 (µ0 ) , first note that every menu B ∈ B has
some action aB such that {a, B} ∈ E, and therefore every connected component Ĝ of the graphhas some action a(Ĝ) in it. Now, for any action a ∈ A, let Ĝa denote its connected component,
and fix some walkaaaa0 B1a aa1 . . . BNa aN afrom a(Ĝa ) to a—so in particular, aa0 = a(Ĝa ) and aaN a = a. Then, let
ua :=

Na 
X



faan ,Bna − faan−1 ,Bna .

n=1
A∪B

Next, for any menu B ∈ B, let λB := uaB − faB ,B . Having constructed (u, λ) ∈ L1 (µ0 )45,The equivalence follows the same reasoning as the characterization of which current functions arise from somevoltage function (e.g., see p. 27 of Bollobás, 2012). For ease of exposition, and because that result is not stated forvector-valued currents, we provide here a self-contained proof in present notation.

55

we need only show it satisfies (11). To that end, consider any edge {a, B} ∈ E, and let ã := aB .
Because they share a neighbor by construction, a and ã belong to the same connected component,and soã ã ãããaa
χ := aa0 B1a aa1 . . . BNa aN a B aN ã BN ã . . . a1 B1 a0
is a testable cycle. We can therefore apply equation (4) to χ to learn
0 =

Na 
X



faan ,Bna − faan−1 ,Bna + (fã,B − fa,B ) +

Nã 
X
n=1

n=1

faãn−1 ,Bnã − faãn ,Bnã= ua + (fã,B − fa,B ) + (−uã )(ua − fa,B ) − λB ,

confirming (11).

We briefly note that some of the content of Corollary 1 is readily recovered beyond the case of
conditionally full-support data sets, but not all of it.
Fact 4. Suppose C satisfies Assumptions A2(i) and A2(ii), and let D = (B, β) be a data set suchB
that C is iteratively differentiable at pβ for each B ∈ B.
1. Given any cycle basis, conditions (ii) and (iii) of Corollary 1 are equivalent.
2. Suppose D is fully mixed. If D is consistent, it satisfies condition (ii) of Corollary 1.
3. Suppose D has full support. If D satisfies condition (ii) of Corollary 1, it is consistent.
Moreover, neither of the latter two implications has a converse in general.
See Appendix C for a proof—and for an example of a fully-mixed, inconsistent data set satisfying the cycle equation; and an example of a full-support, consistent one not satisfying the equation.
Our next goal is proving Proposition 4. To this end, we first prove that under Assumptions A1
and A2, some consistent data set always exists which is linearly independent for every menu.
Lemma 22. Suppose A1(ii) and A2(i) hold. Then for any B ⊆ A, some consistent data set D =(B, β) is such that β B is linearly independent (over B) for all B ∈ B.
Proof. Because µ0 has full support and Ω is metrizable, Assumption A1(ii) (i.e., |A| ≥ |Ω|) implies
existence of a partition {Ωa }a∈A of Ω into |A| measurable non-null sets. For η ∈ R++ , define theutility u via uηa = η1Ωa . Define the set ΩB = ∪a∈B Ωa for any B ⊆ A. For any B ∈ B, take tB ∈ S
146
to be tBa = 1Ωa + |B| 1ΩA\B .
46

If we weakened Assumption A2(i) to assume only that every simple and fully mixed information policy has afinite cost, then the same result would hold with a slightly modified construction.

56

Fix any menu B ∈ B, and for any u ∈ U let
VB (u) := max [E[u · s] − κ(s)] .
s∈SB

Note,
VB (uη ) ≥ E[uη · tB ] − κ(tB ) = ηµ0 (ΩB ) − κ(tB ).
Because κ(tB ) < ∞ by Assumption A2(i), we obtain that
1 B ηV (u ) ≥ µ0 (ΩB ).
lim
inf
η→∞ η
Hence, for sufficiently large η, any SCR sB that is uη -rationalizable over B has
1E[1Ωa · sa ] > µ0 (Ωa )2for all a ∈ B.

So fix η large enough that the above inequality holds for all B ∈ B; and for any B ∈ B, set β B
to be some SCR that is uη -rationalizable over B.
To complete the proof, we argue β B is linearly independent for all B—that is, that {βaB }a∈B
consists of |B| linearly-independent vectors. For this purpose, define the |B| × |B| matrix,
B×B
M := (E[1Ωa · βbB ])(a,b)∈B×B ∈ R+.

Notice M is strictly diagonally dominant, and so has full rank. To complete the lemma, observe
P
that if ζ := (ζa )a∈B ∈ RB is such that a ζa βaB = 0, then M ζ = 0, meaning ζ = 0. We conclude
β B is linearly independent for all B ∈ B.

We now proceed to prove Proposition 4. For a proof sketch, consider the graph generated
by some full-support data set with menu set B, and let X be a set of testable cycles that does
not contain a cycle basis of that graph. The key to the proof is to use the data set generated by
Lemma 22 and the utility function u that rationalizes it for the construction of a new data set
that is inconsistent, but satisfies the cycle equation for all cycles in X . The construction is basedon pairing every menu B with an SCR that is rationalized over B by a slight perturbation of u,where we choose this perturbation so that it satisfies two properties. First, the perturbation is smallenough so that the rationalized SCR is conditionally full support. Second, the perturbation results
in a collection of utility functions whose differences across menus satisfy the cycle equation for
all cycles in X , but fails that equation for some test cycle outside said set. Using Proposition 3 toconnect these utility differences to the cost function’s derivative concludes the proof.
Proof of Proposition 4. Fix some nonempty B ⊆ A, and let G = (V, E) be the graph induced by57

any full-support data set with menu set B. Take X to be a set of testable cycles in G that does notcontain a cycle basis. Our goal is to find a conditionally full-support data set that is inconsistent,
but satisfies (4) for all cycles in X . In what follows, we will invoke various supporting results fromour paper, applying them to versions of our model with action set B ∈ B.
We first argue that some z ∈ RE exists such that `χ · z = 0 for all testable cycles χ ∈ X ,but `χ̌ · z , 0 for some testable cycle χ̌ < X . To find such a z, let L be the vector subspace ofRE spanned by the set of all cycle vectors, and let L̃ be the span of {`χ : χ ∈ X }. Obviously,L̃ ⊆ L ⊆ RE . Moreover, X not containing a cycle basis implies L̃ , L. Therefore, because abasis for L̃ can be completed to one for L, a nonzero linear map from L to R exists that is zero on
L̃. Then, because a basis for L can be completed to a basis for RE , this linear map can be extended
to a linear map from RE to R. Representing this linear map via ` 7→ z · ` (because E is finite) givesa z as desired.
By Lemma 22, a consistent data set D = (B, β) such that β B is linearly independent (hence
full support) for every B ∈ B. Let u0 be some utility that simultaneously rationalizes β B over B
for all B ∈ B.
For every B ∈ B and every  > 0, define uB, ∈ U via
u0 + za,B

if a ∈ B,u 0otherwise,

uB,
a = 

a
a

and take sB, to be some SCR that is uB, -rationalizable over B. We argue that sB, has conditionally full support (over B) for sufficiently small . To that end, define the proper, weak*-lower
semicontinuous, and convex (by Lemma 2) cost function,

κB (s) =



κ(s)

if s ∈ SB ,∞otherwise,

and take
VB : U → R,
u 7→ max E[u · s] − κB (s)s∈S

to be the value function with cost function κB . Letting ∂VB be the subdifferential of VB , Lemma 7
and the definition of κB imply∂VB (u) = argmaxs∈S [E[u · s] − κB (s)] = argmaxs∈SB [E[u · s] − κ(s)] .

58

Thus, sB, ∈ ∂VB (uB, ). Applying Proposition 9, we therefore obtain that ∂VB (u0 ) = {β B }—inparticular, ∂VB is singleton valued at u0 . Since ∂VB is also norm-to-weak* upper hemicontinuousw∗(Lemma 6), that uB, → u0 implies sB, −→ β B . In particular, sB, has full support over B forsufficiently small  > 0. Applying Lemma 20, it follows that sB, is conditionally full support (onB) for all  > 0 small enough.
Pick  > 0 so that sB, is conditionally full support on B for all B ∈ B. Define the data set
D̃ = (B, β̃) by setting β̃ B = sB, . We now conclude the proof by showing D̃ satisfies (4) for all
cycles in X , but violates said equation for the testable cycle χ̌ identified at the beginning of thisproof. For any B ∈ A, Proposition 3 delivers a λB ∈ L1 (µ0 ) such that for all a ∈ B,D̃uB,a = λB + fa,B ,and soD̃
fa,B= u0a − λB + za,B 1.

Thus, given a testable cycle χ = a0 B1 a1 . . . BN aN ,N
X(fan ,Bn −fan−1 ,Bn ) =

n=1

" N
X

#

(u0an − u0an−1 + λBn − λBn )
n=1

+

" N
X

#

(zan ,Bn − zan−1 ,Bn ) = (`χ ·z)1.

n=1

By choice of z, it follows that (4) holds for all cycles χ ∈ X but fails for χ = χ̌, as required.



B. Supplement to Section 7
This appendix provides formal support for any nontrivial claims made in section 7.

B.1. Partial Knowledge of Benefits
In this section, we give a characterization of which SCRs are V-rationalizable for certain wellbehaved instances of V ⊆ U.
Begin with the case in which the relevant set is a singleton, V = {u}. Recall, as explained in
the main text, and shown in Lemma 7, to answer the question of whether or not u rationalizes s is
the same as checking whether or not u is a subgradient of κ at s. The latter condition admits a test
in terms of κ’s directional derivative. Specifically, the SCR s is u-rationalizable if and only if00d+s κ(s ) ≥ E [u · (s − s)]for all s0 ∈ S. One can interpret the above condition as saying the agent does not benefit at the59margin from shifting her behavior from s in any direction.
Our next result generalizes the above test for the case in which V is convex and weakly compact.47 Our generalization replaces the change in the agent’s expected benefits E [u · (s0 − s)] withthe support function of V,
σV : S → R ∪ {±∞},
t 7→ sup E [u · t] .
u∈VGiven a pair s, s0 of SCRs, the quantity −σV (s − s0 ) = inf u∈V E [u · (s0 − s)] gives the lowest
possible marginal increase in the agent’s objective from shifting her behavior from s towards s0 .
Geometrically, the support function describes a closed convex set via its supporting hyperplanes.
The next proposition shows one can test whether s is V-rationalizable by comparing the support
function of V to the cost’s directional derivative at s.
Proposition 10. If V ⊂ U is nonempty, weakly compact, and convex, then s ∈ S κ is V-rationalizable
if and only if every s0 ∈ S κ has
00d+s κ(s ) ≥ −σV (s − s ).

Proof. First, suppose s is V-rationalizable, that is, rationalized by some u ∈ V. Taking any s0 ∈0000S κ , Lemma 7 then implies d+s κ(s ) + E [u · (s − s )] ≥ 0. But observe σV (s − s ) ≥ E [u · (s − s )]
0
0
48
0
0
+
by definition of σV . Hence, d+s κ(s ) + σV (s − s ) ≥ ds κ(s ) + E [u · (s − s )] ≥ 0.
00Conversely, suppose every s0 ∈ S κ has d+s κ(s ) + σV (s − s ) ≥ 0. Toward showing s is Vrationalizable, observe convexity of κ implies every  ∈ [0, 1] and s0 ∈ S have 1 [κ(s + (s0 − s)) − κ(s)] ≤00κ(s0 ) − κ(s); and so taking limits yields d+s κ(s ) ≤ κ(s ) − κ(s). Using this inequality givesn00
0 ≤ 0infκ d+s κ(s ) + σV (s − s )

o

s ∈S

≤ 0infκ {κ(s0 ) − κ(s) + σV (s − s0 )}
s ∈S



0

0



= 0infκ κ(s ) − κ(s) + max E [u · (s − s )]
s ∈S

u∈V

= 0infκ max {κ(s0 ) − κ(s) + E [u · (s − s0 )]}
s ∈S

u∈V

= max 0infκ {κ(s0 ) − κ(s) + E [u · (s − s0 )]} ,u∈V s ∈Sn
A sequence of utilities (un )∞n=1 converges weakly to u if E [u · t] → E [u · t] for all t ∈ S. The set of utilities V is weakly compact if it is weakly closed and uniformly integrable; see Bogachev (2007) Definition 4.5.1 andTheorem 4.7.18.
48
As the proof makes clear, convexity and compactness of V play no role in this direction of the equivalence.
47

60

where the first equality follows from weak compactness of V, and the last equality then follows
from Sion’s minmax theorem (Sion, 1958). Therefore, some u ∈ V exists such that κ(s0 ) − κ(s) +E [u · (s − s0 )] ≥ 0 for all s0 ∈ S κ —that is, s is u-rationalizable.
Proposition 10 is most useful if one can compute directional derivatives of κ and the support
function of the utility set V. Naturally, the tractability of these computations will depend on theform of κ (hence, C) and of V.
When C is differentiable at the information policy revealed by a given SCR, Lemma 13 tells
us we can replace κ with its posterior-separable approximation, and so one need only compute
directional derivatives of the latter.
Corollary 4. If V ⊂ U is nonempty, weakly compact, and convex, and c is a derivative of C ats ∈ S κ , then s is V-rationalizable if and only if every s0 ∈ S κ has
00d+s κc (s ) ≥ −σV (s − s ).

In this case, Lemma 14 provides an explicit formula for the relevant directional derivatives, andLemma 16 provides an even more explicit formula for the same given iterative differentiability.
Below, we provide several examples of sets of utilities to which Proposition 10 can be applied,and explicitly compute the relevant support functions.
Example 7 (Utility bounds). Given constant payoff bound ū ∈ R++ , consider the set49
V := {u ∈ U : |ua | ≤ ū1 ∀a ∈ A}.
Let us demonstrate that V satisfies the hypotheses of Proposition 10 and has σV (t) = ūktk1 forevery t ∈ S. As our analysis shows, computing support functions can in fact be a useful step inverifying the compactness condition on V required by the theorem. With this work in hand, we canapply Proposition 10: A given SCR s ∈ S κ is rationalized by some objective with payoffs bounded00
by ū if and only if every s0 ∈ S κ has d+s κ(s ) ≥ −ūks − s k1 .
To see V satisfies the proposition’s hypotheses, let L+ ⊆ L1 (µ0 ) denote the set of nonnegative
functions. Because V = [(L+ − ū1) ∩ (ū1 − L+ )]A , it follows V is convex and weakly closed. It
remains to compute the support function of V and verify V is weakly compact. To that end, observe
the support function of V evaluated at any t ∈ S takes the formσV (t) = supXu∈V a∈A

E [ua ta ] =XE [ū sign(ta )ta ] = ūa∈AX

E|ta | = ūktk1 ,a∈AIn a standard abuse, we say an element of L1 (µ0 ) is nonnegative if some version of it is nonnegative. We adopt ananalogous abuse for pointwise function inequalities, or (when Ω is ordered) in calling a member of L1 (µ0 ) increasing.
49

61

and (as noted in the second equality above) the program defining this support function attains amaximum at (ū sign(ta ))a∈A . Hence, James’ theorem (Theorem 6.36 in Aliprantis and Border,
2006) implies that V is weakly compact.
Example 8 (Restricted incremental utility). Suppose A = {a0 , . . . , an } for distinct a0 , . . . , an , and
let Fi ⊆ L1 (µ0 ) be nonempty, convex, and weakly compact for each i ∈ {1, . . . , n}. LetnoV := u ∈ U : uai − uai−1 ∈ Fi ∀i ∈ {1, . . . , n} .
Although the unbounded set V is not weakly compact, we can still use Proposition 10. To see wecan, let V0 := {u ∈ V : ua0 = 0}. Because every u ∈ V rationalizes the same set of SCRs as
utility u0 := (ua − ua0 )a∈A ∈ V0 , it follows that a given SCR is V-rationalizable if and only if it
is V0 -rationalizable. Moreover, we show below that V0 satisfies Proposition 10’s hypotheses andcompute the support function of V0 in terms of the support functions of {Fi }ni=1 —demonstrating
useful algebraic properties of support functions that make them easier to compute. The upshot isthat some u ∈ V rationalizes a given SCR s ∈ S κ if and only if0d+s κ(s ) ≥ −

n
X



σFi 



n
X

(saj − s0aj ) , ∀s0 ∈ S κ .

(12)

j=i

i=1

Let us now verify V0 satisfies Proposition 10’s hypotheses. First, observe V0 is nonempty(containing the zero utility) and obviously convex. To see it is compact, definenΦ : L1 (µ0 )A→ L1 (µ0 ) = U


j
X

fi 

f = (fi )ni=1 7→

i=1.

a=aj ∈A

The linear map Φ is clearly norm-to-norm continuous, hence weak-to-weak continuous. It follows
Q
that the image V0 = Φ( ni=1 Fi ) is weakly compact, and Proposition 10 applies.
Finally, let us see how basic algebraic properties of support functions can be used to makethe computation of σV0 more explicit. Recall the adjoint (also known as the transpose) Φ∗ : S =L∞ (µ0 )A → L∞ (µ0 )n is the unique linear norm-continuous map such that every t ∈ S andQPf ∈ ni=1 Fi satisfy ni=1 E [(Φ∗ t)i fi ] = E [t · Φf ]. Then, each t ∈ S has
σV0 (t) = σΦ(Qni=1QnFi ) (t) = σ

i=1∗Fi (Φ t) =nX

σFi ((Φ∗ t)i ) ,i=1

where the second equality follows directly from the definition of the support function and the adjoint, and the third equality follows from additive separability of linear functions on product spaces.
62

Meanwhile, by direct computation, the functional form Φ∗ t =defining the adjoint, and so
σV0 (t) =

n
X

n
X

σFi 

i=1

n
n
j=i taj i=1 satisfies the equation

P

taj  .

j=i

Therefore, s is V-rationalizable if and only if (12) holds.
Example 9 (Bounded increasing differences). Suppose A = {a0 , . . . , an } ⊂ R for a0 < · · · < anand Ω ⊂ R. Let V denote the set of utilities with (i) weakly increasing differences, and (ii) payoffs
Lipschitz of constant ξ with respect to the chosen action. That is, V is the set of all utilities u ∈ Usuch that every i ∈ {1, . . . , n} has uai − uai−1 weakly increasing with magnitude bounded byūi := ξ(ai − ai−1 ). Below, we verify that this example is an instance of Example 8 with someappropriate specification of the {Fi }ni=1 , and the support function σFi takes the form(

σFi (f ) = ūi max
−
∗
ω ∈Ω

Z
(−∞,ω ∗ )f (ω) µ0 (dω) + |f (ω ∗ )|µ0 (ω ∗ ) +Z(ω ∗ ,∞))f (ω) µ0 (dω) .

(13)

A key component of the argument, which is useful for applying Proposition 10 more generally,is that one can compute a support function of a compact convex set by optimizing only over its
extreme points. Substituting (13) into equation (12) yields an explicit characterization of whether
a given SCR is V-rationalizable.
For example, consider the further specialization to the case in which A = Ω = {0, 1} andξ = 1. In this case, the expression that needs to be nonnegative according to equation (12) isE |s1 − s0 |: s01 (0) > s1 (0) and s01 (1) < s1 (1), ps − ps 0

: otherwise.

0d+s κ(s ) + 

1

1

1

In particular, in this case, the magnitude of the “marginal benefit” component is a distance between s and s0 . If s0 matches the state less than s does in both states, then it is the probability thats and s0 choose different actions; and otherwise it is the difference in wholesale action frequencies
between s and s0 .
Let us now formally verify the relevant compactness property and the support function computations. First, in Example 7, we established that {fi ∈ L1 (µ0 ) : |fi | ≤ ūi } was weakly compact.
Moreover, the set of fi ∈ L1 (µ0 ) with weakly increasing version is weakly closed. Hence, thenonempty convex set
Fi := {fi ∈ L1 (µ0 ) : fi weakly increasing, |fi | ≤ ūi },is weakly compact for every i ∈ {1, . . . , n}. Therefore, the computations of Example 8 apply
63directly. Toward a more explicit characterization, let us compute σFi (f ) for each i ∈ {1, . . . , n}and f ∈ L∞ (µ0 ). Because Fi is convex and compact, the Bauer maximum principle says thatevery continuous linear function evaluated over Fi attains its maximum at the extreme points,
ext Fi = {fi ∈ L1 (µ0 ) : fi weakly increasing, |fi | = ūi }.50 Each extreme point therefore takes asimple cutoff form, taking value −ūi to the left of the cutoff and value ūi to the right of the cutoff,and one of these values at the cutoff (if it is a µ0 -atom). Hence,(σFi (f ) = ūi max
−
∗
ω ∈Ω

Z

∗

(−∞,ω ∗ )∗f (ω) µ0 (dω) + |f (ω )|µ0 (ω ) +)Z(ω ∗ ,∞)f (ω) µ0 (dω) .

Finally, consider the specialization to the case in which A = Ω = {0, 1} and ξ = 1. Directcomputation then shows the relevant support function takes the form
σV0 (t) = σF1 (t1 )= max{−E[t1 ], −µ0 (0)t1 (0) + µ0 (1)t1 (1), E[t1 ]}= max {|E[t1 ]| , −µ0 (0)t1 (0) + µ0 (1)t1 (1)}E |t1 |= : t1 (0) < 0 < t1 (1)|E[t1 ]| : otherwise,

as described above.
Example 10 (Action payments). Given u ∈ U and payment bound w̄ ≥ 0, let
V := {u + (wa 1)a∈A : w ∈ [0, w̄]A },which is (being the convex hull of finitely many points) obviously nonempty, convex, and weakly
compact. We can interpret this example as settling a question of implementability with transfers: If
agent has latent utility u, and a principal has the ability to offer action-dependent payments (withpreferences quasilinear in money) bounded by w̄, can SCR s ∈ S κ be incentivized? As we show
below, each s, s0 ∈ S haveσV (s − s0 ) = E [u · (s − s0 )] + w̄2

X0

psa − psa .

a∈A

So for a given SCR s ∈ S κ , letw̄∗ :=50

inf
0

s0 ∈S κ : psa ,psa ∃a∈A2
ps −psa0 |a∈A | aPhi

0E [u · (s0 − s)] − d+s κ(s ) .

The proof that the extreme points take this form is essentially identical to that of Lemma 2.7 from Börgers (2015).

64

So, Proposition 10 implies s ∈ S κ is rationalizable with monetary incentives in [0, w̄] if and only
if w̄ ≥ w̄∗ .
Now, let us verify the described computation. To that end, note each t ∈ S has
σV (t) = E [u · t] + σV−u (t)= E [u · t] + σw̄[co{0,1}]A (t)= E [u · t] + w̄X

σco{0,1} (ta )a∈A= E [u · t] + w̄X

(E [ta ])+ ;a∈Awhereas any s, s0 ∈ S haveX

(E [sa − s0a ])+ =X0psa − psa


+

a∈A

a∈A

=

Xh 
1
2

0

0



psa − psa + 12 psa − psa

i

a∈A

"

=

1
2

=

1
2

X

0
psa − psa



#

+ 12

a∈A

0

psa − psaa∈A0
psa − psa

X

X

.

a∈A

The aforementioned form for the support function follows directly.

B.2. Partial Knowledge of Costs
For the LLR cost function, direct computation shows that any data set data set D = (B, β) andstate ω have
(
D
fa,B
(ω) =

X
ω0

B

θω,ω0
µβ (ω)
θω0 ,ω β B 0Bµβa (ω 0 ) + log βaB−µ (ω ) .
µ0 (ω)µ0 (ω 0 ) aµa (ω 0 )"#)We can express Corollary 1 more succinctly in the language of vectors and matrices. SpecifD
ically, view θ as a matrix in RΩ×Ω , view µ0 as a vector in RΩ , view f D = [fa,B(ω)]ω,{a,B} andΩ×EDµ = [µa,B (ω)]ω,{a,B} as a matrices in R
, and let log µ denote the matrix whose entries are thelogarithms of the corresponding entries of µ. Finally, we find it convenient to reparameterize thecost function, by letting51
η := [diag(µ0 )]−1 [θ − diag(θ1Ω )] ,51

Here, diag : RΩ → RΩ×Ω is the natural embedding taking each vector to a diagonal matrix.

65

a matrix each of whose rows sums to zero.52 It follows readily thatf D = (η − η > )µ − η(log µ).
Corollary 1 then tells us that D is consistent if and only if every testable cycle χ has(η − η > )µ`χ = η(log µ)`χ ,a condition amounting to |Ω|ξ linear restrictions on η (hence on θ), where ξ is the cardinality ofa cycle basis of D as calculated in (3).
Now, let us specialize to the example data set described by (2). The computations that show thisdata set is consistent with mutual information costs in fact show (log µ)`χ is zero for every testable
cycle in a cycle basis, and the uniform prior makes η − η > proportional to θ − θ > . Hence, the data
set is consistent with LLR costs of parameter θ if and only if (θ − θ > )µ`χ is zero for both testable
cycles χ ∈ {χ1 , χ2 } in our cycle basis. This condition clearly holds for every symmetric matrix
θ, and an easy computation shows no nonzero skew-symmetric matrix has zero product with µ`χfor both of these testable cycles χ. Because every square matrix is the sum of a symmetric andskew-symmetric matrix, it follows that the data set is consistent if and only if θ is symmetric.

B.3. Unique Rationalizability and Strict Convexity
The following lemma shows that, when C is strictly convex, the agent’s optimal information choice
is unique, and the only scope for multiple best responses is the willingness to mix over her action
conditional on a realized signal.
Lemma 23. Suppose C is strictly convex, and let u ∈ U. If s, t ∈ S are u-rationalizable, thenps = pt and, for every a ∈ A, the functions sa and ta are proportional.
Proof. First, we show no two u-rationalizable SCRs can generate different information policies.
1
2
To that end, suppose s1 , s2 ∈ S κ have ps , ps ; we show they cannot both be u-rationalizable.
P
P
i
Letting s := 12 2i=1 si , Lemma 5(ii) tells us 21 2i=1 ps  ps , so that
κ (s) = C (ps ) ≤ C

1
2

2
X

p

si

!

< 12

i=1

2
X
i=1



C ps

i



= 12

2
X

 

κ si .

i=1

Hence, E [u · s] − κ(s) > 12 2i=1 [E [u · si ] − κ(si )] ≥ mini=1,2 [E [u · si ] − κ(si )]. Therefore, s
witnesses that at least one of {s1 , s2 } is not u-rationalizable.
P

These matrices are in bijective correspondence with zero-diagonal θ ∈ RΩ×Ω . Indeed, direct computation showsθ = [diag(µ0 )] (η − ι η), where ι ∈ RΩ×Ω is the identity matrix, and is the Hadamard product.
52

66

Now, let s, t ∈ S κ be u-rationalizable. Because r 7→ E [u · r] − κ(r) is concave (by Lemma 2),
it follows that r := 21 (s + t) is u-rationalizable as well. By the above analysis, ps = pt = pr ; inparticular, pr ⊀ 21 ps + 12 pt . Lemma 5(iii) then implies no a ∈ supp(s) ∩ supp(t) exists such that
µsa , µta . Said differently, sa and ta are proportional for every a ∈ A.
Proof of Proposition 5. Let S 1 denote the set of SCRs s ∈ S κ with supp(s) = A and the |A|beliefs {µsa }a∈A all distinct. Below, we show S 1 is weak* open and norm dense in S κ . Before
doing so, let us see this result would deliver the proposition. To that end, let R denote the setof rationalizable SCRs if Ω is infinite, and let it denote the relative interior of S κ if Ω is finite.
Given Proposition 1, every SCR in R is rationalizable, and R is norm dense in S κ . Moreover,by construction, R is open in S κ if Ω is finite. Hence, the intersection R ∩ S 1 is norm dense in
S κ and open in it if Ω is finite. Moreover, if we establish that any u ∈ U and u-rationalizables ∈ R ∩ S 1 are such that s is uniquely u-rationalizable, Lemma 8 would tell us the SCRs in R ∩ S 1
are rationalized by an open set of utilities.53 Finally, let us see that any u ∈ U and u-rationalizables ∈ R ∩ S 1 are such that s is uniquely u-rationalizable. To do so, take any u-rationalizable SCR
t. That s ∈ S 1 implies ps has support size |A|. By Lemma 23, we know pt = ps , and so pt has
support size |A| too. Hence, supp(s) = supp(t) = A. Lemma 23 then also tells us µsa = µta
for every a ∈ A. But then, because {µsa }a∈A are |A| distinct beliefs, it follows from ps = pt that
psa = pta for every a ∈ A. Hence, t = s, as desired.
All that remains is to show S 1 is weak* open and norm dense in S κ . To that end, define the set
S + :=\{s ∈ S κ : E [sa ] > 0}a∈A

and, for each distinct a, b ∈ A, the set
S a,b :=[{s ∈ S κ : E [sa ] E [1Ω̂ sb ] , E [sb ] E [1Ω̂ sa ]} .

Ω̂⊆Ω Borel

By construction, these sets are all weak* open (hence, norm open) in S κ , and so too is the finiteTintersection S 1 = S + ∩ a,b∈A distinct S a,b . Moreover, because a finite intersection of open anddense sets is dense, norm denseness of S 1 will follow if we can show S + is norm dense in S κ andeach pair of distinct a, b ∈ A has S + ∩ S a,b norm dense in S + . To see S + is norm dense, note+1s+ := ( |A|1)a∈A ∈ S κ because ps = δµ0 and C(δµ0 ) < ∞. Because S κ is convex by Lemma 2, it
follows that every s ∈ S κ and  ∈ (0, 1) have (1 − )s + s+ ∈ S κ , which witnesses (taking  → 0)
the SCR s as a norm limit from S + .
Now, fix any pair of distinct a, b ∈ A. It remains to show S + ∩ S a,b is norm dense in S + .
53

Just as in the proof of Theorem 1, we can apply Lemma 8 to the weak*-open set G (resp. R ∩ G) if Ω is infinite(resp. finite).

67

To that end, let s ∈ S + be arbitrary; we want to show s is in the norm closure of S + ∩ S a,b . Ifs ∈ S a,b , we have nothing to show, so we focus on the complementary case in which µsa = µsb =: µ.
Below, we locate an SCR s∗ ∈ S κ such that any proper convex combination of s and s∗ lies in S a,b .
Observe such proper convex combinations necessarily live in S κ (because κ is convex) and so in S +(because); and they can approximate s arbitrarily well by choosing sufficiently skewed weights.
Hence, finding such an s∗ will yield the required denseness property. To locate such an s∗ , weseparately address the case in which µ , µ0 and the case in which µ = µ0 .
First, suppose µ , µ0 , and let s∗ denote the unique SCR with s∗a = 1. That C(δµ0 ) < ∞implies s∗ ∈ S κ . Moreover, any proper convex combination t of s and s∗ is in S a,b because it has
tb = µ, whereas ta is a proper convex combination of µ and µ0 . Thus, s∗ is as required.
Finally, suppose µ = µ0 . By hypothesis, some p ∈ P C exists such that p , δµ0 . Because p
has barycenter µ0 , the distribution p must be nondegenerate. Pooling different posteriors generated
by p if necessary (which will weakly reduce costs and so remain in P C because C is monotone),
we may assume without loss p has binary support {µ∗a , µ∗b }. Note µ0 lies strictly between µ∗a∗∗
and µ∗b . Then, define s∗ to be the unique SCR with µsa = µ∗a and µsb = µ∗b . By construction,∗
κ(s∗ ) = C(ps ) = C(p) < ∞; that is, s∗ ∈ S κ . Finally, any proper convex combination t of s
and s∗ is in S a,b because it has ta and tb being proper convex combinations of µ0 with µ∗a and µ∗b ,respectively. Thus, s∗ is as required.
Proof of Proposition 6. Assumption A2(i) implies, given |Ω| > 1, that P C , {δµ0 }. Hence, Proposition 5 delivers a norm-dense subset of S κ comprising only uniquely rationalizable SCRs. The
result then follows directly from Lemma 21.


B.4. Subdifferentiability, Rationalizability, and Posterior Separable Costs
Proposition 11. Fix some s ∈ S such that ps ∈ feas C, and suppose c is a derivative of C at ps . If∂c(µsa ) is nonempty for all a ∈ supp s, then ∂κ(s) is nonempty, and so s is rationalizable.
Proof. By Lemma 13, it suffices to show s is rationalizable according to κc , which is (by Lemma 7)00equivalent to some u ∈ U satisfying, for every s0 ∈ S κ , the inequality d+s κc (s ) ≥ E [u · (s − s)].
For each a ∈ supp(s), take some fa ∈ ∂c(µsa ). Shifting fa by a constant if necessary—whichR
clearly preserves fa ∈ ∂c(µsa )—assume without loss that fa (ω) µsa (dω) = c(µsa ). Let us seeu ∈ U given by

ua =


f a

: a ∈ supp(s)min c(∆Ω)1: a ∈ A \ supp(s)68yields the desired payoff ranking. Indeed, for any s0 ∈ S, Lemma 16 tells us
0d+s κc (s ) ≥E [(s0a − sa )ua ] +

X

X

h

0

E (s0a − sa ) c(µsa )ia∈supp(s0 )\supp(s)a∈supp(s)0≥ E [(s − s) · u] ,as required.

B.5. Convexity and Monotonicity
The following lemma constructs a “convex monotone envelope” of an information cost function Ĉand establishes some regularity properties of the same.
Lemma 24. Suppose Ĉ : P → R ∪ {∞} is proper and lower semicontinuous. Then,C : P → R ∪ {∞}
p 7→
Q∈∆P:

min
R

Z

Ĉ(q) Q(dq)q Q(dq)p

is well defined (i.e., the minimum exists), proper, lower semicontinuous, convex, and monotone.
Proof. Given the HLPBSSC theorem, the relation  is continuous (i.e., a closed subset of P 2 ). We
can therefore apply a version of the maximum theorem (e.g., Lemma 17.30 from Aliprantis andBorder, 2006) because the barycenter map is continuous and Ĉ is lower semicontinuous: the map
C : P → R ∪ {∞} is well defined and lower semicontinuous.54 Moreover, C is monotone because is transitive, and it is proper because Ĉ is proper and C ≤ Ĉ.
Finally, we turn to convexity. The HLPBSSC theorem implies  is a convex subset of P 2 , andR
so the correspondence P ⇒ ∆P given by p 7→ {Q ∈ ∆P : q Q(dq)  p} has a convex graph.
Hence, C is a (weakly) convex function.
Proof of Proposition 7. Let C be the function defined in the statement of Lemma 24, which (bythat lemma) satisfies all of our standing assumptions on information cost functions.
R
Take any SCR s. Lemma 1 implies a given Q ∈ ∆P can induce s if and only if p Q(dp)  ps .
R
Hence, by Lemma 24, some Q ∈ ∆P of minimum average cost Ĉ(p) Q(dp) can induce s, andthis cost is exactly equal to C(ps ). The result follows.

54

The map takes values in R ∪ {∞} rather than R, but the cited lemma can be applied because R ∪ {∞} ishomeomorphic to a subset of R via a strictly increasing transformation.

69

B.6. Continuous Choice with Bounded Utilities
We begin with an abstract result on V-rationalizability for the case in which V is a well-behaved
linear subspace of U.
Lemma 25. Suppose V ⊆ U is a linear subspace, (S̃, k · k) is a normed space with S̃ ⊇ S, and{ϕ|S : ϕ : S̃ → R linear continuous} is the set of maps S → R given by s 7→ E [u · s] for u ∈ V.
Then, an SCR s is V-rationalizable if and only if
κ(s0 ) − κ(s)> −∞.
s0 ∈S : s ,s
ks0 − skinfκ0Proof. Fix s ∈ S. Viewing κ as a function on S̃ (by letting it take value ∞ on S̃ \ S), thesubdifferential of κ at s takes the formno∂S̃ κ(s) = ϕ ∈ S̃ ∗ : κ(s̃) ≥ κ(s) + ϕ(s̃ − s) ∀s̃ ∈ S̃ .
By hypothesis, {ϕ|S : ϕ ∈ S̃ ∗ } is the set of functions S → R that are given by s 7→ E [u · s] forsome u ∈ V. Hence,∂S̃ κ(s) , ∅⇐⇒{ϕ|S : ϕ ∈ ∂S̃ κ(s)} , ∅⇐⇒∃u ∈ V such that ∀s0 ∈ S, κ(s0 ) ≥ κ(s) + E [u · (s0 − s)]⇐⇒s is V-rationalizable.

Therefore, the lemma follows immediately from Gale’s (1967) duality theorem.

A

Proof of Proposition 8. Letting S̃ = L1 (µ0 ) , whose dual space is naturally identified with V =L∞ (µ0 )A , the result follows directly from the first part of Lemma 25.

Although we have applied Lemma 25 to determine which SCRs are rationalized by somebounded utility, one can vary the norm (generating distinct bounded-steepness conditions) to characterize rationalizability with respect to different classes of utilities. For example, κ exhibits
bounded steepness with respect to the L2 (µ0 ) norm at exactly the SCRs that can be rationalized
by a finite-variance utility. Similarly, bounded steepness of κ with respect to the KantorovichRubinstein norm characterizes rationalizability by a Lipschitz utility.
In contrast to the previous examples, Lemma 25 cannot be applied directly to the case in which
V is the space of continuous functions Ω → R. The reason is that, whereas the above examples
70relied on viewing S as a subset of some S̃ whose dual was naturally identified with V, the space ofcontinuous functions is typically not a dual. Indeed, if Ω is infinite with finitely many connectedcomponents (e.g., if it is [0, 1]), one can easily show (via the Banach-Alaoglu theorem) the spaceof continuous functions is not the dual of any normed space.

B.7. Costly Stochastic Choice
Let us state the analogue of Theorem 2 that we reported in section 7.
Proposition 12. Suppose κ̃ is strictly convex on its domain S κ̃ and is finite and differentiable atevery conditionally full-support SCR. Further, suppose every full-support s ∈ S κ̃ that does not0have conditionally full support admits some s0 ∈ S such that d+s κ̃(s ) = −∞. Then, SCRs yielding
unique subset predictions are weak* dense in S κ̃ .
We now build up to the proof of the above proposition.
Remark 2. In the analysis of this section, we apply Lemmas 4, 6, and 7 and Proposition 1 toκ̃. All of these results were proven under the hypothesis that κ is proper, convex, and weak*
lower semicontinuous (established in Lemma 2 for κ and directly assumed for κ̃). In particular,
inspection of the proofs of these results shows they do not use the fact that κ is derived from C, andso the results can be applied to κ̃ without change.
The next lemma adapts Lemma 19 to the simpler setting in which κ̃ is assumed differentiable.
Lemma 26. Let s ∈ S κ̃ have supp(s) = A, and suppose u is a derivative of κ̃ at s. The following
are equivalent for v ∈ U:(i) SCR s is v-rationalizable; that is, s ∈ argmaxt∈S [E [v · t] − κ̃(t)].
A(ii) Some λ ∈ L1 (µ0 ) and γ ∈ L1 (µ0 )+ exist such that every a ∈ A has
va = λ − γa + ua ,sa γa = 0.
Proof. Let us := v − u ∈ U. Lemma 7 tells us s is v-rationalizable if and only if every s0 ∈ S κ has
0
0
+
0
0
d+
s κ̃(s ) ≥ E [v · (s − s)]. But, that u is a derivative of κ̃ at s implies ds κ̃(s ) = E [u · (s − s)].
We can therefore write the optimality condition as s ∈ argmaxs0 ∈S E [us · s0 ]. As in the proof ofA
Lemma 19, this condition is equivalent to requiring some λ ∈ L1 (µ0 ) and γ ∈ L1 (µ0 )+ to have
us = (λ − γa )a∈A and (γa sa )a∈A = 0.

The following analogue of Corollary 3 is an immediate consequence of Lemma 26.
71

Corollary 5. Let the SCR s have conditionally full support. If κ̃ is differentiable at s, and u and u0
both rationalize s, some λ ∈ L1 (µ0 ) exists such that ua = u0a + λ ∀a ∈ A.
Next, we record an analogue of Lemma 20.
0
Lemma 27. If s ∈ S κ̃ admits a s0 ∈ S such that d+s κ̃(s ) = −∞, then s is not rationalizable.

Proof. For every  ∈ (0, 1), let t = s + (s0 − s). Every u ∈ U then has
11
&0{E [u · t ] − κ̃(t ) − [E [u · s] − κ̃(s)]} = E [u · (s0 − s)] − [κ̃(t ) − κ̃(s)] −−→ −∞.
Thus, for sufficiently small  ∈ (0, 1), the SCR t strictly outperforms s.

Next, we provide an analogue of Lemma 21.
Lemma 28. Suppose κ̃ is finite and differentiable at every conditionally full-support SCR, every0full-support s ∈ S κ̃ without conditionally full support admits some s0 ∈ S such that d+s κ̃(s ) =−∞, and some norm-dense subset S1 of S κ̃ exists that comprises only uniquely rationalizable
SCRs. Then, the set of SCRs yielding unique subset predictions is weak* dense in S.
Proof. The proof of Lemma 21 (which invokes Lemmas 4, 6, and 7 and Proposition 1) can beapplied nearly verbatim, with three minor differences. First, the step invoking Assumption A2(i) is
replaced with the hypothesis that the set S κ̃ contains all conditionally full-support SCRs, because
the latter set is dense in S. Second, we invoke Lemma 27 instead of Lemma 20. Third, we invoke
Corollary 5 instead of Corollary 3.
Finally, we can prove the main result of this section.
Proof of Proposition 12. By Proposition 1 and strict convexity of κ̃, the set S1 of uniquely rationalizable SCRs is norm dense in S κ̃ . The proposition then follows from Lemma 28.
72

C. Auxiliary Material
C.1. Omitted Proofs for Auxiliary Results
This section provides proofs of additional results stated in the previous appendices without proof.
Proof of Corollary 2. Say A = {0, 1} without loss, and note s ∈ S generates state-independent
behavior if and only if ps = δµ0 . We therefore require that if s ∈ S is optimal and ps , δµ0 , thens is uniquely optimal. But ps , δµ0 for ps ∈ P implies both that ps0 , ps1 > 0 and that µs0 , µs1 are
distinct, and hence affinely independent. Proposition 9 therefore applies directly.
∗
Letting p∗ := ps for some u-rationalizable SCR s∗ , the above argument tells us ps = p∗ forevery u-rationalizable SCR s. Therefore (in light of Lemma 1), C being strictly monotone means
every optimal strategy entails information policy exactly p∗ , as required.
Proof of Fact 1. Toward establishing convexity of c, fix any µ1 , µ2 ∈ ∆Ω and τ ∈ (0, 1), and letµ = (1−τ )µ1 +τ µ2 . Let (µn1 , µn2 )n∈N be a sequence of pairs of simply drawn posteriors converging
to (µ1 , µ2 ), which exists by Lipnowski and Mathevet’s (2018) Lemma 2. By definition of a simply
drawn posterior, some sequence (ζ1n , ζ2n )n∈N of pairs from R++ exists such that ζin µni ≤ µ0 for alli ∈ {1, 2} and n ∈ N.
Now, consider any n ∈ N. Take ζ n := 21 min{ζ1n , ζ2n } and µn := (1 − τ )µn1 + τ µn2 for n ∈ N.
−ζ n µn
Observe 2ζ n µn ≤ (1 − τ )ζ1n µn1 + τ ζ2n µn2 ≤ µ0 , and so µ̂n = µ01−ζ
is a well-defined simplyndrawn posterior. Therefore, bothhipn := ζ n δµn + (1 − ζ n )δµ̂n and p̃n := ζ n (1 − τ )δµn1 + τ δµn2 + (1 − ζ n )δµ̂n
are in P F . Because p ∈ feas C, it follows that p + (pn − p) and p + (p̃n − p) are both in P C
for sufficiently small  > 0. Hence, because p̃n  pn , we have (because C is monotone)1[C(p + (p̃n − p)) − C(p + (pn − p))]
&0 
(
)
i h
i
1 h
= lim infC(p + (p̃n − p)) − C(p) − C(p + (pn − p)) − C(p)&0 

0 ≤ lim inf=

Zc(µ) (p̃n − p)(dµ) −Zc(µ) (pn − p)(dµ)

hi= ζ n (1 − τ )c(µn1 ) + τ c(µn2 ) − c(µn ) .
Finally, that (1−τ )c(µn1 )+τ c(µn2 ) ≥ c(µn ) for every n ∈ N yields (1−τ )c(µ1 )+τ c(µ2 ) ≥ c(µ)
because c is continuous. The lemma follows.


73

Proof of Fact 2. Let f ∈ L1 (µ0 ) be the difference of two derivatives of c ∈ C at the simplyR
drawn µ.55 By hypothesis, every simple µ̃ ∈ ∆Ω has f (ω) (µ̃ − µ)(dω) = 0, and hence,RRf (ω) µ̃(dω) = f¯ := f (ω) µ(dω).
R
Because f (ω) µ0 (dω) = f¯, the event Ω̂ := {f ≥ f¯} ⊆ Ω has µ0 (Ω̂) > 0. Let µ̃ := (µ0 |Ω̂)
denote the conditional measure, which is a simply drawn posterior. Therefore,
0 = µ0 (Ω̂)

Z h

i

h

i

f (ω) − f¯ µ̃(dω) = E 1Ω̂ (f − f¯) .

An expectation of a nonnegative random variable can be zero only if the random variable is almost
surely zero, yielding two consequences. First, 1Ω̂ (f − f¯) = 0 ∈ L1 (µ0 ), so that f ≤ f¯. Second,that f ≤ f¯ and E [f ] = f¯ implies f = f¯ ∈ L1 (µ0 ), a constant.
Finally, that every derivative of c at µ has µ-expectation c(µ) then implies f¯ = 0.56
Proof of Fact 3. Every q ∈ P has q  δµ0 , and so p + (q − p)  p + (δµ0 − p) for any  ∈ (0, 1).
Hence,
inf q∈P C , ∈(0,1)=

inf ∈(0,1)=

lim&01[C(p + (q − p)) − C(p)]
1[C(p + (δµ0 − p)) − C(p)] (by monotonicity)1[C(p + (δµ0 − p)) − C(p)] (by convexity),

which is equal to d+p C(δµ0 ) by definition.

Proof of Fact 4. To see the first numbered fact, note that the proof of the equivalence of conditions (ii) and (iii) in Corollary 1 makes no use of conditional full support.
To see the second fact, observe that the proof of Corollary 1 applies directly (mutatis mutandis)
to a generalized version of a data set in which the set of menus is a multiset of menus. In particular,
it applies to the generalized data set D0 in which each menu B ∈ B is replaced with the smaller
menu suppβ B ; note that (relabeling vertices in B in the obvious way) the graph GD0 is the sameas the graph GD , giving rise to the same testable cycles. Because the data set D is consistent, so
is D0 . That D is fully mixed implies D0 has conditionally full support, and so Corollary 1 tells us
condition (ii) is satisfied for D0 , hence (because the graphs are the same) for D.
To see the third fact, note Proposition 3 tells us a sufficient condition for the support-B SCR β B
Dto be u-rationalizable over B ∈ B is that some λB ∈ L1 (µ0 ) has ua = λB + fa,B
for every a ∈ A.
55

Note that the proof relies only on the “Newton quotient” property of ∇cµ , together with its average value normalization, and so does not require that c ∈ C.
R
56
If we did not require the normalization that ∇cµ (ω) µ(dω) = c(µ)—which would not affect the definition ofdifferentiability—the derivative would be unique only up to addition of a constant function.

74

Otherwise following verbatim the proof that condition (ii) implies condition (i) in Corollary 1
delivers the third fact.
Now, we pursue the “moreover” part. We will provide examples showing the second and third
facts do not generally have converses.
Consider the case in which A = {1, 2, 3}, B = {{1, 2}, A}, Ω = {0, 1}, β {1,2} has conditionally full support with β {1,2} (0) , β {1,2} (1), β1A (0) = β1A (1) = 1, and C is strictly monotone—e.g.,
C could be mutual information as in Example 1. This data set is fully mixed and vacuously satisfies
condition (ii) of Corollary 1 because the graph has no nontrivial cycles. Let us observe the data set
is not consistent. The reason is that, if it were u-rationalizable for u ∈ U, then β {1,2} and β A would
both be u-rationalizable over {1, 2}. This conclusion would contradict Proposition 9 (applied tothe model with menu {1, 2}), which would tell us β {1,2} is uniquely u-rationalizable over {1, 2}.
Consider now the case in which A = {0, 12 , 1}, B = {{0, 12 }, {0, 1}, { 12 , 1}}, Ω = {0, 1},BB
βmin
B (0) = βmax B (1) = 1 for every B ∈ B, and C has derivative c at full information suchthat c : ∆Ω → R is globally differentiable and not affine—e.g., C could be given by C(p) =Rµ(1)2 p(dµ). The data set clearly has full support. Identifying ∆Ω with [0, 1] in the obvious
way, direct computation shows every µ ∈ [0, 1] has ∇cµ (ω) = c(µ) + (ω − µ)c0 (µ) for ω ∈DΩ. Because µB
a = 1a=max B for a ∈ B ∈ B, we can then write fa,B = c(1a=max B ) + (ω −DD(0) = c0 (1a=max B ). The testable(1) − fa,B
1a=max B )c0 (1a=max B ), which in particular implies fa,Bcycle χ = 0 {0, 21 } 12 { 21 , 1} 1 {0, 1} 0 therefore has`χ · f D (1) − `χ · f D (0) = [c0 (1) − c0 (0)] + [c0 (1) − c0 (0)] + [c0 (0) − c0 (1)] = c0 (1) − c0 (0) , 0.
In particular, `χ · f D , 0, so that condition (ii) of Corollary 1 does not hold. To conclude, we
observe that D is consistent. Indeed, because c is Lipschitz, direct computation shows D is ratio
nalized by uθ ∈ U given by uθa (ω) := −θ(a − ω)2 for large enough θ ∈ R.

C.2. On Inducible Belief Distributions
In this section, we validate footnote 5’s claim that belief distributions and signal structures areequivalent formalisms. To state the equivalence, let us invest in some terminology.
Definition 1. Given a Polish space M , a signal is a measurable map τ : Ω → ∆M , and a belief
map is a measurable map π : M → ∆Ω. For a given pair (τ, π) of such maps,• Say the pair (τ, π) is Bayes consistent if every Ω̂ ⊆ Ω and M̂ ⊆ M haveZΩ̂τ (M̂ |ω) µ0 (dω) =

Z Z
M̂

75

π(Ω̂|m) τ (dm|ω) µ0 (dω).

• Say the pair (τ, π) generates p ∈ ∆∆Ω if every B ⊆ ∆Ω hasZτ π −1 (B) | ω µ0 (dω) = p(B).

Note every signal admits some Bayes-consistent belief map, and that updated beliefs are almost
surely unique, and hence unique in distribution. These observations amount to recording a standard
disintegration result in present notation.
Fact 5. Suppose M is Polish and that τ is a signal. Define P ∈ ∆(Ω × M ) by lettingP(Ω̂ × M̂ ) :=ZΩ̂

τ (M̂ |ω) µ0 (dω)

for every measurable Ω̂ ⊆ Ω and M̂ ⊆ M . Define PM := margM P.
(i) Some belief map π exists such that the pair (τ, π) is Bayes consistent.
(ii) If belief maps π, π̃ are such that both (τ, π) and (τ, π̃) are Bayes consistent, π(Ω̂|m) =π̃(Ω̂|m) for PM -almost every m.
(iii) Suppose belief maps π, π̃ are such both (τ, π) and (τ, π̃) are Bayes consistent; and supposep, p̃ ∈ ∆∆Ω are such that (τ, π) and (τ, π̃) generate p and p̃, respectively. Then, p = p̃.
Proof. Given a belief map π, note the pair (τ, π) is Bayes consistent if and only if P(Ω̂ × M̂ ) =M̂ π(Ω̂|m) PM (dm) for every measurable Ω̂ ⊆ Ω and M̂ ⊆ M . Hence, (i) and (ii) both follow
directly from the disintegration theorem (Kallenberg, 2017, Theorem 1.23).57 Finally, any two(∆Ω-valued) random variables on a probability space that agree almost surely must have the same
distribution; hence, (iii) follows directly from (ii).
R

The following result shows every signal generates (when pairing it with belief updating to makeit Bayes consistent) a unique belief distribution, and that a belief distribution can be generated inthis way if and only if it averages to the prior. This classic result—whose proof we include for thesake of completeness—is again a straightforward consequence of the disintegration theorem.
Fact 6. Let M be uncountable and Polish, and let p ∈ ∆∆Ω. The following are equivalent:58(i) Some Bayes-consistent pair (τ, π) generates p.
(ii) Some signal τ is such that every Bayes-consistent (τ, π) generates p.
57

By (Aliprantis and Border, 2006, Theorem 19.7), probability kernels as in Kallenberg (2017) coincide with measurable maps into the space of measures.
58
As the proof demonstrates, (i) and (ii) are equivalent and imply (iii), even if M is not assumed uncountable.
Moreover, the proof (along with the proof of Fact 5) applies verbatim to Polish, non-compact Ω.

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(iii) The distribution p averages to µ0 , that is, p ∈ P.59
Proof. Given any signal τ , Fact 5 tells us some Bayes-consistent pair includes τ and that no twosuch pairs generate distinct belief distributions. Hence, (i) is equivalent to (ii). In what follows, weshow (i) is equivalent to (iii).
To see (i) implies (iii), suppose (τ, π) is Bayes consistent and generates p. Toward (iii), consider any measurable Ω̂ ⊆ Ω. Applying Bayes consistency with M̂ = M yields µ0 (Ω̂) =RR
π(Ω̂|m) τ (dm|ω) µ0 (dω); and because (τ, π) generates p, any bounded integrable f : ∆Ω → R
RRR
hasf (π(m)) τ (dm|ω) µ0 (dω) = f (µ) p(dµ). Hence, using f (µ) := µ(Ω̂) gives µ0 (Ω̂) =Rµ(Ω̂) p(dµ), delivering (iii).
Conversely, suppose (iii) holds, from which we establish (i). By the Borel isomorphism theorem (Srivastava, 2008, Theorem 3.3.13), some bimeasurable surjection π : M → ∆Ω exists.60
R
Now, define the measure P ∈ ∆(Ω × M ) by letting P(Ω̂ × M̂ ) := π(M̂ ) µ(Ω̂) p(dµ) for every
measurable Ω̂ ⊆ Ω and M̂ ⊆ M . By (iii) and because π is surjective, the marginal of P on its firstcoordinate is µ0 . Hence, the disintegration theorem (Kallenberg, 2017, Theorem 1.23) delivers
Rsome measurable τ : Ω → ∆M such that P(Ω̂ × M̂ ) = Ω̂ τ (M̂ |ω) µ0 (dω) for every measurableΩ̂ ⊆ Ω and M̂ ⊆ M . Let us see that (τ, π) witnesses (i). Indeed, for any measurable Ω̂ ⊆ Ω andM̂ ⊆ M , combining the disintegration property defining τ with the definition of P impliesZΩ̂τ (M̂ |ω) µ0 (dω) =Zµ(Ω̂) p(dµ).

(14)π(M̂ )

Specializing equation (14) to the case of Ω̂ = Ω and M̂ = π −1 (B) for some measurable B ⊆∆Ω tells us, because π is surjective, that (τ, π) generates p. All that remains now is to verify(τ, π) is Bayes consistent. To show it is, define the map Pm ∈ ∆M by letting PM ∈ ∆M :=R
τ (M̂ |ω) µ0 (dω) for every measurable M̂ ⊆ M . We know (τ, π) generates p; that is, PM ◦ π −1 =p. Hence, every measurable Ω̂ ⊆ Ω and M̂ ⊆ M haveZ Z
M̂π(Ω̂|m) τ (dm|ω) µ0 (dω) =ZM̂

π(Ω̂|m) PM (dm) =Bayes consistency then follows from equation (14).

Zµ(Ω̂) p(dµ).

π(M̂ )

R
Recall elements of P are those p ∈ ∆∆Ω with barycenter µ0 or, equivalently, with µ(Ω̂) p(dµ) = µ0 (Ω̂) for
every measurable Ω̂ ⊆ Ω.
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Moreover, in the special case in which M ⊇ ∆Ω, we can take π with π(µ) = µ for each µ ∈ ∆Ω, generating awitnessing (τ, π) in which each signal realization from ∆Ω leads to itself as the updated belief.
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