Nonseparable Costly Information Acquisition and Revealed Preference

Transcription

1 Nonseparable Costly Information Acquisition and Revealed Preference Christopher P. Chambers, Ce Liu, and John Rehbeck Department of Economics University of California-San Diego Last Updated: July 12, 2017 Abstract We provide revealed preference characterizations for choices made under costly information acquisition. In particular, we examine nonseparable, multiplicative, and constrained costly information acquisition. The revealed preference characterization is performed using primitives from Caplin and Dean (2015). Each model has an analogous model in the literature on decisions under uncertainty. The techniques we use parallel the duality properties in the standard consumer problem. We thank Mark Dean and Pietro Ortoleva for useful comments and suggestions. 1

2 1 Introduction We study a model of costly information acquisition in the spirit of Caplin and Dean (2015). Our innovation is to characterize a model when the cost of information acquisition may not be additively separable. Instead, we impose that preferences are increasing in the gross expected utility derived from stochastic choices and probability updating follows Bayes law. The most salient reason for studying nonseparable costs is to incorporate wealth, or utility, effects. For example, if utility is measured in terms of wealth, then a nonseparable cost of information reflects the fact that individuals at different wealth levels may have different incentives to acquire information. This flexibility allows an individual who is well-off to have a weaker incentive to acquire information. This behavior is induced by a higher cost of acquiring information when an individual has a higher wealth level. As special cases of the model, we characterize a representation with multiplicative costs of information and a representation with a constrained information set. Multiplicative cost is relevant, for example, when information costs a fixed share of the decision maker s gross payoff. This work builds on the recent contribution of Caplin and Dean (2015) using the same environment and primitives. Caplin and Dean (2015) provides a revealed preference test for costly information acquisition when the costs are additively separable from gross expected utility. The environment from Caplin and Dean (2015) considers a decision maker facing actions with statecontingent payoffs. The decision maker chooses an information structure and then makes stochastic choices conditioned on the signal received from the information structure. If a researcher is able to observe a decision maker s stochastic choices conditioned on the true state of the world, then there is a natural revealed information structure from the decision maker s stochastic choices. Caplin and Dean (2015) establish that an acyclicity condition on 2

3 the revealed information structure and an optimality condition on stochastic choices are necessary and sufficient for stochastic choice data to be consistent with the hypothesis of optimal costly information acquisition. Our primary characterization generalizes the acyclicity condition of Caplin and Dean (2015) in the same way that the generalized axiom of revealed preference (see Houthakker (1950); Richter (1966); Chambers and Echenique (2016)) generalizes the cyclic monotonicity condition of Rockafellar (1966) or the condition of Koopmans and Beckmann (1957). 1 Of note is that the generalized axiom of revealed preference is ordinal. Our proofs exploit a duality between direct and indirect utility which has recently been fruitfully studied by Chateauneuf and Faro (2009) and Cerreia- Vioglio, Maccheroni, Marinacci and Montrucchio (2011a,b). Roughly, in classical consumer theory, the maximization of any direct utility function subject to fixed budget and price gives rise to a quasiconcave and monotone indirect utility function. In fact, any quasiconcave and monotone function can also be written as the dual of a direct utility function in the above fashion. In our framework, the decision maker s preference for nonseparable costly information acquisition is given by a quasiconcave and monotone function, which would correspond to the indirect utility function in classical consumer theory. Let us now explain our results in more detail. The contribution of Caplin and Dean (2015) is, roughly, to characterize a decision maker by their gross expected utility from experimentation less a cost of the experiment. An experiment, or signal, is a probability distribution over posteriors (as in Blackwell (1953)). Mathematically, up to a normalization, probability measures and normalized price vectors are the same object. In this sense, if we treat the expected utility of experimentation as wealth and the probability distri- 1 See also Brown and Calsamiglia (2007) and Chambers, Echenique and Saito (2016) for variants of this condition in an explicit revealed preference framework. 3

4 bution over posteriors as prices, we are dealing with an object that closely resembles an indirect utility function in classical consumer theory. This resemblance can in fact be made formal. The indirect utility that is considered in Caplin and Dean (2015) is one for which there are no wealth effects. Using the indirect utility, we can equivalently represent such a decision maker by their direct utility function. When there are no wealth effects, there are well-known revealed preference tests for rationalizability. These are based on the cyclic monotonicity condition of Rockafellar (1966). We go a step further, and observe that a general indirect utility function (that is, one in which wealth effects can be present) has an direct utility representation. In this case, the test for rationalizability by a utility function is well-known: the generalized axiom of revealed preference. The only distinction between what we are doing here and the classical theory is that here, the direct utility is increasing and convex, whereas the indirect utility is quasiconcave. This owes to the fact that the direct utility here is derived from the indirect utility via maximization (whereas in the standard case, it results from minimization). With this approach, results which are otherwise nontrivial to obtain fall directly out of the representation. For example, it is well-known that an indirect utility function has a quasiconcave representation; this is almost by definition. So, the fact that our nonlinear aggregator can be taken to be quasiconcave is without loss. Similarly, by taking this approach, it is relatively simple to show that the cost function in the Caplin and Dean (2015) setup can be taken to be convex without loss. As we see it, the main objective of this paper is to establish simple characterizations of costly information acquisition models by leveraging existing revealed preference and duality techniques in a unified approach. As such, the paper should also serve as a didactic exercise. This paper also establishes a link between the literature on costly information acquisition and 4

5 decisions under uncertainty. In particular, the nonseparable model corresponds to Cerreia-Vioglio et al. (2011b), the multiplicative model corresponds to Chateauneuf and Faro (2009), the additive separable cost corresponds to Maccheroni, Marinacci and Rustichini (2006), while the constrained information set model corresponds to Gilboa and Schmeidler (1989). This paper is related to other works on costly information acquisition, decision making with an unknown state of the world, and revealed preference. Costly information acquisition has received study from a various perspectives in Denti, Mihm, de Oliveira, Ozbek et al. (2016), Ellis (2016), and Matejka and McKay (2014). The theory is also related to models when the information state of a decision maker is unknown e.g. Masatlioglu, Nakajima and Ozbay (2012) and Manzini and Mariotti (2014). Boundedly rational behavior has been studied with revealed preference conditions in Fudenberg, Iijima and Strzalecki (2015), Aguiar (2016), and Allen and Rehbeck (2016). The paper proceeds as follows. Section 2 presents the notation and some useful facts. Section 3 introduces and characterizes the nonseparable generalization of Caplin and Dean (2015). Section 4 presents a variant of the model whereby choice of information structure is costless, but is constrained to lie in some unknown set. Section 5 presents a model with a multiplicative cost of information acquisition. Section 6 relates the conditions to those in Caplin and Dean (2015), provides examples that clarify behavior allowed by the various models, provides guidance on out of sample prediction, and discusses some limitations. Finally, Section 7 concludes. Proofs are relegated to the appendix. 5

6 2 Preliminaries and Revealed Information Structures 2.1 Notation We study a decision maker facing actions with state-contingent payoffs. 2 Notation is consistent with Caplin and Dean (2015) whenever possible for ease of comparison. We study a variety of models that are increasing in gross expected utility and satisfy Bayes law. A decision maker chooses actions whose outcome depends on a finite number of states of the world. Let Ω denote a finite set of states. Let X denote a set of outcomes. Therefore, the set of all actions (state-contingent outcomes) is X Ω. The set of all finite decision problems is given by A = {A X Ω A < }. As in Caplin and Dean (2015) we investigate when a researcher has a state dependent stochastic choice dataset from decision problems in A. For A A, (A) refers to the set of probability distributions over actions in A. Definition 1 A state dependent stochastic choice dataset is a finite collection of decision problems D A and related set of state dependent stochastic choice functions P = {P A } A D where P A : Ω (A). Denote the probability of choosing an action a conditional on state ω in decision problem A as P A (a ω). We assume that the prior beliefs of the decision maker µ Γ = (Ω) are known. Moreover, we assume that the utility index u : X R is a known function. We take an abstract approach to modeling the choice of an information structure. Each subjective signal is identified with it s associated posterior beliefs γ Γ. Thus, an information structure is given by a finite support 2 The ideas discussed here are broader if one considers general mappings over posteriors. 6

7 distribution over Γ that satisfies Bayes law. Definition 2 The set of information structures, Π, comprises all Borel probability distributions over Γ, π (Γ), that have finite support and satisfy Bayes law. A distribution over posteriors satisfies Bayes law if the distribution over posteriors is a mean-preserving spread of the prior µ denoted as E π [γ] = γ Supp(π) where π(γ) = Pr(γ π) = ω Ω µ(ω)π(γ ω). γ π(γ) = µ We now provide definitions necessary to discuss gross expected utility. Given a utility index, each decision problem A A induces a posterior value function, f A : Γ R, which maps posterior beliefs γ to the maximal utility from A under posterior γ. Formally, for any decision problem A and posterior belief γ f A (γ) = max a A γ(ω)u(a(ω)). ω Ω Definition 3 We denote the gross expected utility induced by an information structure π Π as π f A = γ Supp(π) where π(γ) = Pr(γ π) = ω Ω µ(ω)π(γ ω). π(γ)f A (γ) This representation of gross expected utility is intuitive since f A is a continuous function on Γ and the set of continuous functions on Γ is topologically dual to the set of countably additive Borel measures on Γ (Aliprantis and Border (2006), Theorem 14.15). 7

8 2.2 Revealed Information Structures While we present several models of costly information acquisition, the analysis relies on the recovery of a revealed information structure from the state dependent stochastic choice data. Using the procedure from Caplin and Dean (2015), we associate each chosen action to a subjective information state. The revealed information structure may not be identical to the true information structure. However, the revealed information structure is a garbling (as defined in Blackwell (1953)) of the true information structure. The relationship between the true information structures and revealed information structures allows us to order the information structures and deduce conditions on revealed information. Without further delay, we define revealed posteriors and revealed information structures. Definition 4 Given µ Γ, A D, P A P, and a Supp(P A ), the revealed posterior γ A a Γ is defined as γ a A(ω) = Pr(ω a is chosen from A) = µ(ω)p A (a ω) ν Ω µ(ν)p A(a ν). Definition 5 Given µ Γ, A D, and P A P, the revealed information structure π A Π is defined by π A (γ ω) = {a Supp(P A ) γ a A =γ} P A (a ω) and induces a revealed distribution on posteriors π A such that π A (γ) = ω Ω µ(ω) π A (γ ω). The revealed information structure for decision problem A is a finite probability measure over the revealed posteriors. As mentioned before, we use the notion of garbling to partially order information structures. 8

9 Definition 6 The information structure π Π (with posteriors γ j ) is a garbling of ρ Π (with posteriors η i ) if there exists a Supp(ρ) Supp(π) matrix B with non-negative entries such that for all i {1,..., Supp(ρ) } we have γ j Supp(π) bi,j = 1 and for all γ j Supp(π) and ω Ω that π(γ j ω) = b i,j ρ(η i ω). η i Supp(ρ) In other words, π a garbling of ρ if there is a stochastic matrix B that can be applied to ρ that yields π. We present two important properties about garblings that we use extensively in the analysis of different models of costly information acquisition. Lemma 1 For π Π and P A P, we say π is consistent with P A if there exists a choice function C A : Supp(π) (A) such that for all γ Supp(π), C A (a γ) > 0 γ(ω)u(a(ω)) γ(ω)u(b(ω)) for all b A ω Ω ω Ω and for all ω Ω and a A P A (a ω) = π(γ ω)c A (a γ). γ Supp(π) If π is consistent with P A, then π A is a garbling of π. This lemma only depends on the definition of f A and is proved in Caplin and Dean (2015). The lemma says that if an information structure is consistent with the state dependent stochastic choice dataset, then the revealed information structure is a garbling. The models we examine all use the same definition f A and so this property holds throughout the rest of the analysis. The second property of garblings is Blackwell s theorem (Blackwell, 1953). Blackwell s theorem establishes the notion that some information structures are more valuable than others. In particular, if π is a garbling of ρ, then ρ yields weakly higher gross expected utility in any decision problem. 9

10 Remark 1 Given a decision problem A A and π, ρ Π with π a garbling of ρ, then ρ f A π f A. Using these two properties, we can make statements regarding the value of the gross expected utility at the revealed information structure. For example, for all decision problems A, B D if π A is an information structure consistent with choice data P A, then the gross expected utility satisfies f B π A f B π A. Lastly, we have that f A π A = f A π A since the two information structures induce the same state dependent choices. 3 Nonseparable Costly Information We place minimal restrictions on a decision maker s preferences on information structures. The only condition we impose is that preferences are monotone increasing in gross expected utility. Definition 7 Given µ Γ and u : X R, a state dependent stochastic choice dataset (D, P) has a nonseparable costly information representation if there exists a function V : R Π R { }, information structures {π A } A D, and choice functions {C A } A D such that: 1. Monotonicity: For all π Π and for all t, s R, if t < s and V (t, π) >, then V (t, π) < V (s, π). 2. Non-triviality: For all t R, there exists π t Π such that V (t, π t ) >. 3. Information is optimal: For all A D, π A arg max π Π V (π f A, π). 4. Choices are optimal: For all A D, the choice function C A : Supp(π A ) (A) is such that given a A and γ Supp(π A ) with 10

11 C A (a γ) = Pr A (a γ) > 0, then γ(ω)u(a(ω)) γ(ω)u(b(ω)) ω Ω ω Ω for all b A. 5. The data is matched: For all A D, given ω Ω and a A, P A (a ω) = π A (γ ω)c A (a γ). γ Supp(π A ) We now define the properties that completely characterize the model. The first condition is similar to the generalized axiom or revealed preference. Condition 1 (Generalized Axiom of Costly Information (GACI)) We say the dataset (D, P) satisfies GACI if for all sequences ( π A1, f A1 ),..., ( π Ak, f Ak ) with A i D for which π Ai f Ai π Ai f Ai+1 for all i (with addition modulo k), then equality holds throughout. Comparing this condition to GARP, we see that the π play a role similar to prices and the f terms play a role similar to consumption bundles albeit with the inequality reversed. The GACI condition rules out the possibility of cycles in gross expected utility across different decision problems. Using this condition, we invoke a version of Afriat s theorem (see Chambers and Echenique (2016)). Lemma 2 (Afriat s Theorem) Let D be finite. For all (A, B) D 2, let α A,B R. If for all A D one has α A,A = 0 and for any sequence A 1, A 2,..., A k D with α Ai,A i+1 0 (with addition mod k) for all i it follows that α Ai,A i+1 = 0 for all i, then there exist numbers U A and λ A > 0 such that for all (A, B) D 2, U A U B + λ B α B,A. The other condition that characterizes the nonseparable costly information representation is the no improving action switches (NIAS) condition. This condition was first examined in the study of Bayesian decision makers in Caplin and Martin (2015). 11

12 Condition 2 (NIAS) Given µ Γ and u : X R, a dataset (D, P) satisfies NIAS if, for every A D, a Supp(P A ), and b A, µ(ω)p A (a ω)(u(a(ω)) u(b(ω))) 0 ω Ω The combination of GACI and NIAS completely characterizes the model of nonseparable costly information acquisition. Theorem 1 Given µ Γ and u : X R, the dataset (D, P) has a nonseparable costly information representation if and only if it satisfies GACI and NIAS. This shows the basic result that GACI and NIAS are equivalent to the nonseparable costly information representation. However, one can impose additional properties on the nonseparable costly information representation. These conditions are monotonicity, quasiconcavity, and a normalization property on the function V (, ). Condition 3 The function V : R Π R { } satisfies weak monotonicity in information if for any t R and π, ρ Π with π a garbling of ρ, then V (t, ρ) V (t, π). The monotonicity condition says that if one adds noise to a signal ρ, then the noisier signal is cheaper. This is one definition of monotonicity and it agrees with the notion of informativeness introduced in Blackwell (1953). Condition 4 The function V : R Π R { } is quasiconcave if for any (t 1, π 1 ), (t 2, π 2 ) R Π and λ [0, 1], V (λt 1 + (1 λ)t 2, λπ 1 + (1 λ)π 2 ) min{v (t 1, π 1 ), V (t 2, π 2 )}. 12

13 The quasiconcavity condition is similar to the convexity of the cost function in Caplin and Dean (2015). This condition says if we have a mixture of a level of gross expected utilities and information structures, then the utility of the mixture is weakly higher than the worst case of the two environments being mixed. In particular, this implies quasiconcavity in information structures if one sets t 1 = π 1 f and t 2 = π 2 f. Condition 5 Define π 0 as the information structure with π 0 (µ ω) = 1 for all ω Ω. The function V : R Π R { } satisfies the normalization if V (0, π 0 ) = 0. The normalization condition says that utility is normalized to zero when the gross expected utility is zero and an individual does not update their prior. Theorem 2 Given µ Γ and u : X R, the data set (D, P) satisfies GACI and NIAS if and only if it has a nonseparable costly information representation that satisfies Conditions 3, 4 and 5. In the proof, we show that all of these conditions can be established using the construction of V (, ) in Theorem 1. 4 Constrained Costly Information The previous section studies nonseparable costly information acquisition, but there are other structures on preferences that are of interest. Now we consider when an individual is constrained to choose an information structure within some fixed set of information structures. The interpretation is that the decision maker does not have access to the full set of information structures when updating the prior, but the choice of information structure is costless. 13

14 Definition 8 Given µ Γ and u : X R, a state dependent stochastic choice dataset (D, P) has a constrained costly information representation if there exists a set Π c Π of available information structures, information structures {π A } A D, and choice functions {C A } A D such that: 1. Non-triviality: The set Π c. 2. Information is optimal: For all A D, π A arg max π Πc π f A. 3. Choices are optimal: For all A D, the choice function C A : Supp(π A ) (A) is such that given a A and γ Supp(π A ) with C A (a γ) = Pr A (a γ) > 0, then γ(ω)u(a(ω)) γ(ω)u(b(ω)) for all b A. ω Ω ω Ω 4. The data is matched: For all A D, given ω Ω and a A, P A (a ω) = π A (γ ω)c A (a γ). γ Supp(π A ) A constrained costly information structure is characterized by a condition similar to the weak axiom of cost minimization (Varian, 1984). Using this intuition, the revealed information structures are analogous to inputs of production and f A are analogous to prices of inputs. Condition 6 (Weak Axiom of Costly Information (WACI)) The dataset (D, P) satisfies WACI if for all A, B D that π A f A π B f A. Theorem 3 Given µ Γ and u : X R, the dataset (D, P) has a constrained costly information representation if and only if it satisfies WACI and NIAS. 14

15 Similar to the nonseparable case, additional structure can be placed on a constrained costly information representation without restricting observable behavior. Using standard arguments, the constraint set Π c can be made convex. Theorem 4 Given µ Γ and u : X R, the dataset (D, P) has a constrained costly information representation with a convex Π c if and only if it satisfies WACI and NIAS. 5 Multiplicative Cost of Information We now study a multiplicative costly information representation. In this representation, the cost is interpreted as losing a fraction of the gross expected utility. Definition 9 Given µ Γ and u : X R +, a state dependent stochastic choice dataset (D, P) has a multiplicative costly information representation if there exists a function K : Π [0, 1], information structures {π A } A D, and choice functions {C A } A D such that: 1. Non-triviality: There exists π Π such that K(π) < Information is optimal: For all A D, π A arg max π Π [(1 K(π))(π f A )] Choices are optimal: For all A D, the choice function C A : Supp(π A ) (A) is such that given a A and γ Supp(π A ) with 3 We note that the results that follow also hold assuming a function R : Π R + and π A arg max π Π [R(π)(π f A )]. We prefer the (1 K(π)) notation to interpret the costs as a fraction of gross expected utility. 15

16 C A (a γ) = Pr A (a γ) > 0, then γ(ω)u(b(ω)) for all b A. ω Ω γ(ω)u(a(ω)) ω Ω 4. The data is matched: For all A D, given ω Ω and a A, P A (a ω) = π A (γ ω)c A (a γ). γ Supp(π A ) We note that one difference in the statement of the multiplicative costly information representation is that the utility index u is required to be nonnegative. While this is more restrictive than the other cases, this is a common property of multiplicative representations. For example, Chateauneuf and Faro (2009) make such an assumption. The condition that characterizes the multiplicative costly information representation is a version of the homothetic axiom of revealed preference; see Varian (1983). 4 Condition 7 (Homothetic Axiom of Costly Information (HACI)) Given data set (D, P), define D 0 = {A D ω Ω µ(ω)u(a(ω)) = 0 for all a A}. We say the dataset (D, P) satisfies HACI if for all sequences ( π A1, f A1 ),..., ( π Ak, f Ak ) with A i D\D 0, that k π Ai f Ai+1 i=1 π Ai f Ai 1 (with addition modulo k). HACI is essentially the homothetic axiom of revealed preference restricted to decision problems that give positive gross expected utility. The decision problems that give zero gross expected utility are removed since they can be trivially rationalized and they would create an indeterminate fraction. Theorem 5 Given µ Γ and u : X R +, the dataset (D, P) has a multiplicative costly information representation if and only if it satisfies HACI and NIAS. 4 It can also be derived as a relatively easy corollary from the general work of Rochet (1987). 16

17 As in the case of the nonseparable costly information representation, we are able to put additional properties on the function K. We find that K respects monotonicity with respect to the Blackwell partial order, is convex, and satisfies a normalization property. We now define these properties and the give a statement of the theorem. Condition 8 The function K : Π [0, 1] satisfies weak monotonicity in information if ρ, π Π with π a garbling of ρ, K(π) K(ρ). Condition 9 The function K : Π [0, 1] is convex in information structures if for for any π 1, π 2 Π and λ [0, 1], K(λπ 1 + (1 λ)π 2 ) λk(π 1 ) + (1 λ)k(π 2 ). Condition 10 Define π 0 as the information structure with π 0 (µ ω) = 1 for all ω Ω. The function K : Π [0, 1] satisfies normalization if K(π 0 ) = 0. Theorem 6 Given µ and u : X R +, data set (D, P) satisfies HACI and NIAS if and only if it has a multiplicative costly information representation that satisfies Conditions 8, 9, and Relation of Models, Examples, Prediction, and Limitations 6.1 Relationship to Additively Separable Model As a point of reference, we examine how the nonseparable costly information representation relates to the additive costly information representation in Caplin and Dean (2015). First, we provide the definition of an additive costly information representation. 17

18 Definition 10 Given µ Γ and u : X R, a state dependent stochastic choice dataset (D, P) has an additive costly information representation if there exists a function K : Π R { }, information structures {π A } A D, and choice functions {C A } A D such that: 1. Non-triviality: There exists π Π such that K(π) <. 2. Information is optimal: For all A D, π A arg max π Π [π f A K(π)]. 3. Choices are optimal: For all A D, the choice function C A : Supp(π A ) (A) is such that given a A and γ Supp(π A ) with C A (a γ) = Pr A (a γ) > 0, then γ(ω)u(a(ω)) γ(ω)u(b(ω)) for all b A. ω Ω ω Ω 4. The data is matched: For all A D, given ω Ω and a A, P A (a ω) = π A (γ ω)c A (a γ). γ Supp(π A ) Caplin and Dean (2015) showed that an additive costly information representation is characterized by the NIAS condition and a no improving attention cycles (NIAC) condition. The NIAC condition is defined below. Condition 11 (No Improving Attention Cycles (NIAC)) Given µ Γ and u : X R, a dataset (D, P) satisfies NIAC if for all sequences ( π A1, f A1 ),..., ( π Ak, f Ak ) with A i D, then k π Ai f Ai i=1 k π Ai+1 f Ai i=1 where addition of the indices is modulo k. 18

19 have 5 π f A π f A > π f B π f B. The interpretation of NIAC is that one cannot cycle through the information structures and improve the gross expected utility. From the definition of NIAC and GACI, it is easy to see that if a dataset satisfies NIAC, then the dataset also satisfies GACI with equality. Proposition 1 If the dataset (D, P) satisfies NIAC, then it also satisfies GACI. 6.2 Wealth Effects A nonspeparable model of costly information acquisition generalizes the additively separable model by allowing wealth effects. Consider the following example where an individual chooses a non-trivial information structure when utility received from actions is low. However, the individual chooses a less informative information structure (in the Blackwell sense) when facing a menu where actions yield higher utility and there is a higher return to information. This type of decision violates additively separable information costs, however nonseparable information costs can accommodate this behavior. We now define what we mean by higher return to information and give a result on pairs of menus. We provide an explicit numerical example of this behavior in Example 1 in Section 6.3. Definition 11 Menu A provides a higher return of information than menu B if for any information structure π and π a garbling of π with π π, we 5 This definition is non-vacuous. In fact, it can be shown that menu A provides a higher return to information than menu B if and only if f A = f B + g where g is a strictly convex function. 19

20 We establish that an individual with an additive costly information representation can never choose a less informative information structure when faced with a menu that has a higher return of information. Proposition 2 Suppose D = {A, B} for dataset (D, P) with menu A providing a higher return of information than menu B. If π A is a garbling of π B, then the choice data violates NIAC and thus cannot be generated by an additive costly information representation. The next result shows that a nonseparable model accommodates this behavior when the menu that provides a higher return to information also yields higher utility for any posterior. Proposition 3 Suppose D = {A, B} for dataset (D, P) with menu A providing a higher return of information than menu B. If π A is a garbling of π B, NIAS is satisfied, and f A > f 6 B, then this dataset is rationalized by a nonseparable costly information representation. The proof of the Proposition 3 only requires the hypothesis f A > f B. The additional assumptions are in place only to highlight the difference with Proposition 2. This hypothesis is also not necessary for the behavior described in the proposition. For example, as long as the revealed information structures have support points where the difference between optimal actions is strict, then the conclusion of Proposition 3 holds. 6.3 Example Datasets In this section, we examine several simple datasets to distinguish properties of the various models. All of the examples assume that Ω = {ω 1, ω 2 } and the prior is given by µ = ( 1 2, 1 2). We consider menus A = {a, b} and A = {a, b } where the properties of the acts are given in the examples. 6 We say f A > f B if f A (γ) > f B (γ) for all γ Γ. 20

21 The following example is based on an example form Caplin and Dean (2015) that follows the ideas in Proposition 2 and Proposition 3. The example data satisfies NIAS and GACI, but violates NIAC. We also note the dataset satisfies HACI and violates WACI. Example 1 [Less information at higher gross expected utility] Let the actions in menu A and A take the following values: 0 if ω = ω 1 2 if ω = ω 1 u(a(ω)) = u(b(ω)) = 2 if ω = ω 2 0 if ω = ω 2 0 if ω = ω u(a 1 10 if ω = ω (ω)) = u(b 1 (ω)) =. 10 if ω = ω 2 0 if ω = ω 2 Now consider the following choice probabilities given by P A (a ω 1 ) = 2 10 P A (a ω 2 ) = 8 10 P A (a ω 1 ) = 3 10 P A (a ω 2 ) = 7 10 where the choice of b and b are given by the complementary probabilities. These choices generate the following revealed posteriors ( 2 γ A a = 10, 8 ) ( 8 ; γ A b = 10 10, 2 ) 10 ) γ a A = ( 3 10, 7 10 ; γ b A = ( 7 10, 3 10 Each revealed posterior has the same probability of occurring so that ). π A ( γ A) a = π( γ A) b = π( γ A a ) = π( γb A ) = 1 2. The optimal decision rules for these posteriors give f A ( γ A) a = f A ( γ A) b = 1.6 ; f A ( γ A a ) = f A( γ A b ) = 1.4 f A ( γ a A ) = f A ( γb A ) = 7 ; f A ( γa A) = f A ( γ b A) = 8 21

22 One can verify from this information that NIAS holds. Since the decision rules give the same utility for each posterior, we know that π A f A = 1.6 ; π A f A = 1.4 π A f A = 7 ; π A f A = 8. Note that π A f A + π A f A = 8.6 < 9.4 = π A f A + π A f A so that NIAC fails. However, it follows that π A f A < π A f A and π A f A < π A f A so that there are no cycles that violate GACI. Moreover, we find the data satisfies HACI since ( ) ( ) πa f A πa f A = π A f A π A f A ( ) ( ) 14 = 1 70 and the dataset is consistent with a multiplicative cost of information acquisition. Lastly, the dataset violates WACI since π A f A < π A f A. The behavior in Example 1 is consistent with having higher information costs at higher gross expected utility. in each state than menu A. Note menu A gives weakly more utility However, the revealed information structure from A yields a lower gross expected utility with menu A than the revealed information structure from A. This occurs since the information structure π A is more informative that π A. Therefore, it must be that the revealed information structure from A is more costly than that of A at higher gross expected utility. At this point, it is natural to wonder if an individual could instead reveal choosing more information at higher gross expected utility. Example 2 shows that this is the case. For the remaining examples, we suppress calculations 22

23 to more quickly see the implications. While Example 2 does not satisfy HACI, it is easy to create an example such that individuals can choose more informative information and satisfy HACI. Example 2 [More information at higher gross expected utility] Let the actions in menu A and A take the following values: 2 if ω = ω 1 4 if ω = ω 1 u(a(ω)) = u(b(ω)) = 4 if ω = ω 2 1 if ω = ω 2 3 if ω = ω u(a 1 5 if ω = ω (ω)) = u(b 1 (ω)) =. 5 if ω = ω 2 1 if ω = ω 2 Now consider the following choice probabilities given by P A (a ω 1 ) = 4 10 P A (a ω 2 ) = 6 10 P A (a ω 1 ) = 3 10 P A (a ω 2 ) = 7 10 where the choice of b and b are given by the complementary probabilities. One can calculate that NIAS holds and that π A f A = 3 ; π A f A = 4 π A f A = 4.1 ; π A f A = NIAC fails since π A f A + π A f A = 7.1 < 7.25 = π A f A + π A f A. Again there are no cycles that violate GACI since π A f A < π A f A but π A f A < π A f A. The dataset fails HACI since ( ) ( ) πa f A πa f A 1.05 > 1. π A f A π A f A 23

24 Lastly, WACI is violated since π A f A < π A f A. Thus, this data is only rationalized by a nonseparable costly information representation. There are now many examples that satisfy GACI. However, one may wonder what type of data will violate GACI. As GACI is an ordinal version of information acquisition, a violation depends on how information structures interact with the magnitude of utility differences across states. We show in Example 3 a violation of GACI. Example 3 [Violation of GACI] Let the actions in menu A and A take the following values: 2 if ω = ω 1 4 if ω = ω 1 u(a(ω)) = u(b(ω)) = 5 if ω = ω 2 1 if ω = ω 2 5 if ω = ω u(a 1 1 if ω = ω (ω)) = u(b 1 (ω)) =. 2 if ω = ω 2 4 if ω = ω 2 Now consider the following choice probabilities given by P A (a ω 1 ) = 1 10 P A (a ω 2 ) = 8 10 P A (a ω 1 ) = 8 10 P A (a ω 2 ) = 1 10 where the choice of b and b are given by the complementary probabilities. One can calculate that NIAS holds and that π A f A = 4 ; π A f A = 4.1 π A f A = 4 ; π A f A = 4.1. From these numbers, one has that π A f A < π A f A and π A f A < π A f A 24

25 which is a strict cycle so GACI is violated. Thus, this dataset is not rationalized by any model of costly information acquisition. We elucidate why this dataset violates GACI. The difference of utility to make an optimal decision from menu A in state ω 1 is u(b(ω 1 )) u(a(ω 1 )) = 2, and the difference in state ω 2 is u(a(ω 2 )) u(b(ω 2 )) = 4. However, the individual makes choices consistent with an information structure that is more informative when the gain of gross utility is lower. The same is true for menu A with the states and actions reversed. It is clear that this choice violates any utility that is monotone in the gross expected utility. Using the utility numbers of acts from Example 3, it is easy to show an example that satisfies WACI. WACI requires that information structure yield higher utility from a menu when the utility difference is greater. We give an explicit example in Example 4. Example 4 [Dataset that satisfies WACI] Let the actions in menu A and A take the values from Example 3. Now consider the following choice probabilities given by P A (a ω 1 ) = 2 10 P A (a ω 2 ) = 9 10 P A (a ω 1 ) = 9 10 P A (a ω 2 ) = 2 10 where the choice of b and b are given by the complementary probabilities. One can calculate that NIAS holds and that π A f A = 4.1 ; π A f A = 4 π A f A = 4.1 ; π A f A = 4. WACI holds since π A f A > π A f A and π A f A > π A f A. The dataset satisfies GACI, NIAC, and HACI since WACI holds. 25

26 6.4 Out of Sample Prediction Caplin and Dean (2015) provide some suggestions on how the information from a dataset can be used to provide out of sample prediction. They give an example of the restrictions on choice probabilities for a specific two state environment with a uniform prior and menus of two acts. We consider the same case here for GACI. Let the states be given by Ω = {ω 1, ω 2 }. Let the menus be denoted A = {a, b} and A = {a, b }. Assume that u(a(ω 1 )) > u(b(ω 1 )) and u(b(ω 2 )) > u(a(ω 2 )). Similarly, assume that u(a (ω 1 )) > u(b (ω 1 )) and u(b (ω 2 )) > u(a (ω 2 )). From Caplin and Dean (2015), NIAS is equivalent on menu A to u(b(ω 2 )) u(a(ω 2 )) P A (a ω 1 ) max P u(a(ω 1 )) u(b(ω 1 )) A(a ω 2 ), u(b(ω 2 )) u(a(ω 2 )) P u(a(ω 1 )) u(b(ω 1 )) A(a ω 2 ) + u(a(ω 1))+u(a(ω 2 )) u(b(ω 1 )) u(b(ω 2 )) u(a(ω 1 )) u(b(ω 1 )) An equivalent condition holds for menu A. Next assume that the dataset with menus {A, A } satisfies NIAS. Moreover, assume that the decisions are aligned as implicitly assumed in Caplin and Dean (2015). We say the data are aligned if a = arg max c {a,b} b = arg max c {a,b} ω {ω 1,ω 2 } ω {ω 1,ω 2 } γ a A (ω)u(c(ω)), γ b A (ω)u(c(ω)), and similar conditions hold for choices from A using the revealed information structure π A. There is a GACI cycle if. π A f A π A f A and π A f A π A f A with one inequality strict. Under the above assumptions of NIAS and aligned choices, there is a strict cycle if P A (a ω 1 ) 1 + P A (a ω 2 ) 2 β P A (a ω 1 ) 1 + P A (a ω 2 ) 2 β 26

27 where 1 = u(a(ω 1 )) u(a (ω 1 )) + u(b (ω 1 ) u(b(ω 1 )) 2 = u(a(ω 2 )) u(a (ω 2 )) + u(b (ω 2 ) u(b(ω 2 )) β = u(b (ω 1 )) + u(b (ω 2 )) u(b(ω 1 )) u(b(ω 2 )). Therefore, any probabilities that satisfy these inequalities with at least one strict inequality violate a nonseparable costly information representation. In general, suppose one has a menu M A such that M / D. If the dataset D satisfies NIAS and GACI, we can use the information to place bounds on the information structures that are consistent with the model using the restrictions of GACI and NIAS. The full set of restrictions is given by a supporting set as defined in Varian (1984). Denote the set of information structures that support the menu M that are consistent with GACI and NIAS by S GACI (M) = {π M Π {( π A, f A )} A D (π M, f M ) satisfies NIAS and GACI}. This set places restrictions on π M that can be translated to restrictions on individual state dependent stochastic choices. It is easy to define supporting sets for multiplicatively separable, additively separable, and constrained costly information representation. While the supporting set is often difficult to compute, it provides the full set of π M consistent with a given representation. 6.5 Limitations The revealed preference conditions for costly information acquisition often provide interesting bounds and intuition for these models. Moreover, these representations suggest some datasets of particular interest. For example, one notes that an additive costly information representation has the property of being translation invariant in gross expected utility. Similarly, a 27

28 multiplicative costly information representation has the property of being scale invariant in gross expected utility. One may want to look at choices from menus of this type to violate an additively separable or multiplicatively separable costly information representation respectively. However, a dataset with menus that are additive utility translations of one another always satisfy NIAC. Similarly, a dataset with menus that are scale shifts of one another always satisfy HACI. To study these questions, we provide two definitions. For a menu A = {a 1,..., a n } A and c R let A + c = {a 1,..., a n} be the menu that adds a constant utility c to each act. That is, u(a i(ω)) = u(a i (ω)) + c for i = 1,..., n and all ω Ω. Similarly, let ca = {ca 1,..., ca n } be the menu where the utility of all acts is multiplied by c, so u(a i(ω)) = cu(a i (ω)) for i = 1,..., n and all ω Ω. Proposition 4 Let µ Γ and u : X R. If the dataset (D, P) satisfies NIAS, D = {A + c 1, A + c 2,..., A + c M }, and for all m = 1,..., M that c m R, then the dataset is rationalized by the additive costly information representation. Proposition 5 Let µ Γ and u : X R +. Suppose the dataset (D, P) satisfies NIAS, D = {c 1 A, c 2 A,..., c M A}, and and for all m = 1,..., M that c m R +, then the dataset is rationalized by the multiplicative costly information representation. 7 Conclusion In this paper, we provide revealed preference characterizations for several models of costly information acquisition. The characterization of these models follows directly from classical revealed preference theory. We also provide extensive examples showing how the revealed information acquisition differs 28

29 across models. Moreover, the models can be used to generate out of sample predictions, albeit with some difficulty. Appendix A Proofs of Results Proof of Theorem 1. ( ) First, we show that a nonseparable costly information representation satisfies NIAS. Fix A D, π A arg max π Π V (π f A, π), and C A : Supp(π A ) (A) and a Supp(P A ). By definition of a nonseparable costly information representation, we know that the V (π A f A, π A ) is monotone in π A f A and choices are optimal conditional on posteriors. Thus, if a was chosen when γ was realized, then the expected utility must be weakly higher for these γ. For γ such that C A (a γ) > 0, ω Ω γ(ω)u(a(ω)) ω Ω γ(ω)u(b(ω)) b A. The proof now follows from arguments in Caplin and Dean (2015) that are reproduced here for completeness. Recall that γ(ω) = µ(ω)π A(γ ω) ν Ω µ(ν)π(γ ν), which can be substituted on both sides and the denominator cancels so ω Ω µ(ω)π A (γ ω)u(a(ω)) ω Ω µ(ω)π A (γ ω)u(b(ω)) b A. Therefore, [ ] C A (a γ) µ(ω)π A (γ ω)u(a(ω)) γ Supp(π A ) ω Ω [ ] C A (a γ) µ(ω)π A (γ ω)u(b(ω)) γ Supp(π A ) ω Ω b A 29

30 since C A (a γ) are either zero or positive multiples of the earlier introduced inequalities. Next, recall from data matching that P A (a ω) = γ Supp(π A ) π A(γ ω)c A (a γ). Therefore, we see that ω Ω µ(ω)u(a(ω))p A (a ω) = ω Ω = µ(ω)u(a(ω)) γ Supp(π A ) γ Supp(π A ) γ Supp(π A ) π A (γ ω)c A (a γ) [ ] C A (a γ) µ(ω)u(a(ω))π A (γ ω) ω Ω [ ] C A (a γ) µ(ω)u(b(ω))π A (γ ω) ω Ω = ω Ω µ(ω)u(b(ω))p A (a ω) where the first set of equalities follows from substitutions, the inequality follows from optimality conditional on γ, and the last equality follows from the same substitutions above. Rearranging this inequality shows that NIAS is satisfied. Next, we show that a nonseparable costly information representation implies GACI. Observe arg max π V (π f A, π) = V (π A f A, π A ) by definition. We first establish that V (π A f A, π A ) > for all A D. To see this, notice that for all A D, f A is a continuous function on the compact set Γ, so f A achieves a minimum value c A. By non-triviality, there exists πa c such that V (c A, πa c ) >. Observe πc A f A c A. By monotonicity, V (πa c f A, πa c ) V (c A, πa c ) >. Since π A is the optimal choice, we have V (π A f A, π A ) V (c A, πa c ) >. Suppose without loss of generality that π Ai f Ai π Ai f Ai+1 for i = 30

31 {1,..., k} (with addition modulo k). It follows that V (π Ai f Ai, π Ai ) = V ( π Ai f Ai, π Ai ) V ( π Ai f Ai+1, π Ai ) (1) V (π Ai f Ai+1, π Ai ) V (π Ai+1 f Ai+1, π Ai+1 ) Since V ( π Ai f Ai, π Ai ) = V (π Ai f Ai, π Ai ) > for all i, strict monotonicity in the first component of V implies that the inequality in (1) is strict if π Ai f Ai < π Ai f Ai+1. Suppose there is a strict inequality in the sequence, then we obtain the contradiction V (π A1 f A1, π A1 ) < V (π A1 f A1, π A1 ). Consequently, we must have π Ai f Ai = π Ai f Ai+1 for all i. ( ) The converse is a direct application of Afriat s Theorem. Let α A,B = π A (f B f A ) for all (A, B) D 2. Observe that by GACI, the condition in Afriat s Theorem is satisfied. Conclude there is U A and λ A > 0 such that for all (A, B) D 2, U A U B λ B π B (f A f B ). Taking negatives and letting Ũ A = U A, we have Ũ B + λ B π B (f A f B ) ŨA. Most of the remaining construction follows Afriat s theorem directly. Let C(Γ) be the space of continuous functions on Γ. Define U : C(Γ) R by U(f) = max Ũ A + λ A D A π A (f f A ) Clearly, U is convex, continuous, and monotone increasing 7 (as the maximum of a finite number of continuous affine functionals). For every A D, U(f A ) = ŨA by construction. Moreover, for every A D, if π A f π A f A, then U(f) U(f A ), which is also straightforward by construction. 7 The functional U is monotone increasing in f if (f g)(γ) > 0 for all γ Γ, then U(f) > U(g). 31

32 Define V : R Π R { } by V (t, π) = inf f C(Γ) {U(f) : π f t}. Observe that the monotonicity condition is trivially satisfied for fixed π, since a greater t reduces the set of f C(Γ) satisfying the inequality. The assumption that for each t R, there exists a π t Π such that V (t, π t ) > is also satisfied. In fact, we will show V (t, π A ) > for any t R and A D. For any t R, let G t = {g C(Γ) U(g) t}. G t is closed and convex by the continuity and convexity of U( ). Note that C(Γ) is the topological dual to the set of signed Borel measures with bounded variation over Γ (Aliprantis and Border (2006) Theorem 14.15). Let M(Γ) be the set of such measures on Γ. Fix  D. Note that for any f Gt that Ũ + λâ πâ (f fâ) U(f) = max Ũ A + λ A D A π A (f f A ) t. Rearranging the equation gives sup πâ f t Ũ + λâ π  f  f G t λâ Let K(t) = t Ũ +λâ π  f Â. Note that the function K is monotonically increasing with domain and range both spanning the reals. The function K 1 λâ is well-defined and monotonic, with K 1 (x) > for all x R. It follows that G t {f πâ f K(t)}. Note that for all f such that πâ f K(K 1 (t)) it follows that, U(f) Ũ + λ  π  (f f  ) K 1 (t). It follows by definition that for all t R V (t, πâ) = inf {U(f) : πâ f t} K 1 (t) >. f C(Γ) We now assert that for all A D, π A arg max π Π V (π f A, π). First, from the monotonicity property of the U function V ( π A f A, π A ) = inf {U(f) : π A f π A f A } f C(Γ) = U(f A ) 32

33 Second, for any π Π, we have V (π f A, π) = inf f C(Γ) {U(f) : π f π f A } U(f A ), since π f A π f A. Therefore V (π f A, π) V ( π A f A, π A ) for all π Π. Therefore, the revealed information structure is optimal for V. 8 We now show data matching and choices are optimal by following Caplin and Dean (2015) and using NIAS. Next we show that there exists stochastic choice functions {C A : Supp( π A ) (A)} A D that satisfy optimality and matches data. For each γ Supp( π A ), define: P A (a) if γ {b A: γ a C A (a γ) = ba =γ} P A(b) A = γ 0 otherwise where P A (a) = ω Ω P A(a ω)µ(ω) is the unconditional probability of choosing action a from decision problem A. Note the C A (a γ) > 0 only if γ A a = γ. The NIAS condition implies that µ(ω)p A (a ω)u(a(ω)) µ(ω)p A (b ω)u(b(ω)) ω Ω ω Ω γ A(ω)u(a(ω)) a γ A(ω)u(b(ω)) a ω Ω ω Ω The second line follows by dividing both sides by P A (a). Thus, NIAS ensures that the choices are optimal. It remains to show that the data is matched. In other words, P A is generated from the information structure π A and choices C A. First, note 8 We note that a version of Roy s identity holds (Roy (1947)). Observe that by definition of V, if π f A w implies U(f A ) V (w, π). We conclude that π f A π A f A implies U(f A ) V ( π A f A, π). We have already shown that U(f A ) = V ( π A f A, π A ). Thus, if π f A π A f A, then V ( π A f A, π A ) V ( π A f A, π). 33

34 that for any b, b A such that γ A b = γb A, implies that for any ω Ω such that γ A b (ω) > 0, then P A (b ω) P A (b ω) = P A(b) P A (b ). Thus, for every ω Ω and a A such that P A (a) > 0, then π A (γ ω)c A (a γ) = π A ( γ A ω)c a A (a γ A) a γ Supp( π A ) Therefore, the data is matched. Proof of Theorem 2. = P A (c ω) {c A: γ A c = γa A } = P A (c ω) {c A γ A c = γa A } = P A (a ω). P A (a) {b A γ A b = γa A } P A(b) P A (a ω) {b A γ A b = γa A } P A(b ω) Let V : R Π R { } be the function constructed as in the proof of Theorem 1. The function Ṽ (t, π) = V (t, π) V (0, π 0 ) satisfies all the additional properties. First, note that Ṽ (0, π 0) = V (0, π 0 ) V (0, π 0 ) = 0 so the normalization condition is satisfied. Since the difference of V and Ṽ is a constant, we can check quasiconcavity and weak monotonicity of V. Next, we check weak monotonicity. If π is a garbling of ρ, then V (t, ρ) = inf {U(f) ρ f t} f C(Γ) inf {U(f) π f t} f C(γ) = V (t, π) since π f t implies that ρ f t by Remark 1 so the infimum is taken over a weakly smaller set of functions. Thus, weak monotonicity in the second argument of V holds. 34

35 Lastly, we examine quasiconcavity of V. Let (t 1, π 1 ), (t 2, π 2 ) R Π, then for λ [0, 1] V (λt 1 +(1 λ)t 2, λπ 1 +(1 λ)π 2 ) = inf {U(f) λπ 1 f+(1 λ)π 2 f λt 1 +(1 λ)t 2 }. f C(Γ) Note that if λπ 1 f + (1 λ)π 2 f λt 1 + (1 λ)t 2, then either π 1 f t 1 or π 2 f t 2. Therefore, for f C(Γ) we have {f λπ 1 f +(1 λ)π 2 f λt 1 +(1 λ)t 2 } {f π 1 f t 1 } {f π 2 f t 2 }. Therefore, the infimum of U over the first set,v (λt 1 + (1 λ)t 2, λπ 1 + (1 λ)π 2 ), is greater than or equal to the infimum of U over the second set, min{v (t 1, π 1 ), V (t 2, π 2 )}. Thus, quasiconcavity holds. Proof of Theorem 3. We note that NIAS is equivalent to optimal choices and matched data. Therefore, we focus on non-triviality and optimal information. ( ) Suppose the data is represented by a constrained costly information representation and for all A D that π A arg max π Πc π f A. Since the utility depends only on gross expected utility, then π A f A = π A f A π B f A π B f A. The first equality follows from equivalent choices, the next inequality follows from optimality, while the final inequality follows Remark 1. ( ) Suppose WACI holds. Let Π c = A D { π A}. For D nonempty, Π c. 9 Moreover, for any A, B D, we have π A f A π B f A. In other words, for all A D we have π A arg max π Πc π f A. Therefore nontriviality and optimal information hold. Proof of Theorem 4. From the proof of Theorem 3 we only need to prove that WACI implies a convex constraint set. Let conv( Π c ) = 9 If D =, then let Π c = Π. 35