Costly information acquisition

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Abstract

We provide revealed preference characterizations for choices made under various forms of costly information acquisition. We examine nonseparable, multiplicative, and constrained costly information acquisition. In particular, this allows the possibility of unknown time delay for acquiring information. The techniques we use parallel the duality properties in the standard consumer problem.

Introduction

Acquiring information is an integral part of decisions under uncertainty. Most existing research on costly information acquisition studies costs that are additively separable from the expected payoff. This assumes the cost incurred from acquiring information is independent of expected payoff. One can interpret these preferences as an individual having a fixed production technology to acquire information. However, the cost structures of information acquisition can be more complicated.

For example, there may be significant costs incurred from the time delay waiting for information to arrive. Consider when an oil company is deciding between locations to drill for oil. To acquire information, in addition to the monetary expenses to finance geological surveys, there are also significant costs incurred from delayed realization of profits. Suppose the payoff from drilling at a site for each state of the world is given by a net-present-value from the time drilling begins. If the oil sites have a higher net-present-value, then this translates into higher costs incurred through discounting. Importantly, costs from time delay now interact with the expected payoff.

In this paper, we study a general model of costly information acquisition that allows for interactions between the information cost and the expected payoff from the decision problem. Apart from the standard assumptions of expected utility maximization and Bayesian updating, the only additional assumption of the model is that the decision maker prefers higher expected payoffs. As special cases of the model, we characterize a representation with multiplicative costs of information and a representation with a constrained information set.

The paper of Caplin et al. (2015) investigates the particular case of the model studied here when costs are additive. Our model should be viewed as a direct generalization of this contribution. This being said, the work is motivated by classical economic environments. There are several standard economic frameworks in which we would not expect costs to be additive.

The multiplicative cost model is a particularly interesting case. There are several standard economic environments in which we would expect costs to arise multiplicatively, rather than additively. The most compelling environment is when acquiring information results in a time delay. In standard exponential discounting models, delay enters payoffs multiplicatively (e.g. for discount factor δ, utility takes the form δDelay×Expected Payoff). Thus, when information acquisition induces different delays, the behavior cannot be rationalized by a model with an additive cost of information acquisition. For a formal representation, see Example 2.

There are also other examples when the costs are multiplicative. For example, the cost of information acquisition may accrue because of some probability of an absolute breakdown. For example, eliciting the information may involve some type of illegal activity where if the acquirer is caught, then they get nothing. In this case, the probability of not being caught enters multiplicatively into a (risk-neutral) individual's utility.

A more straightforward example involves the individual directly contracting with an outside provider of information, who insists on a profit-sharing agreement with the individual. Here the share of profits asked for can depend on the information sold. In such a setup, the profit sharing obviously enters multiplicatively.

More broadly, the class of nonseparable costly information acquisition models nest behavior generated when there are potentially multiple sources of the cost of information acquisition. For example, the decision maker may incur both an additively separable cost to access an information structure, as well as a discounting cost from time delay.1

Caplin et al. (2015) provide a revealed preference test for costly information acquisition when costs are additively separable. When there is only a cost from discounting, we show that behavior is characterized essentially by the Homothetic Axiom of Revealed Preference (See Varian (1983)).

We take a revealed preference approach that builds on the recent contribution of Caplin et al. (2015). In particular, the model considers a decision maker facing actions with state-contingent payoffs. The decision maker chooses an information structure and makes stochastic choices conditioning on the signal received from the information structure. Using state dependent stochastic choices, there is a natural revealed information structure that facilitates the analysis. Our main result characterizes the general model of costly information acquisition with an axiom on expected payoffs that resembles the Generalized Axiom of Revealed Preference.2 To emphasize the potential interaction between expected utility and cost, we refer to such model as a nonseparable costly information acquisition model.

Our results generalize the No Improving Attention Cycles condition of Caplin et al. (2015) in the same way that the Generalized Axiom of Revealed Preference generalizes the cyclic monotonicity condition of Rockafellar (1966) or the condition of Koopmans and Beckmann (1957).3 Importantly, cyclic monotonicity is equivalent to rationalization via a quasi-linear utility function, which imposes cardinal restrictions on consumption data. Thus, Caplin et al. (2015) reflects a type of cardinal model, while the model here is ordinal. Using the intuition from the consumer problem, we show that data consistent with nonseparable costly information preferences can be taken to satisfy quasiconcavity and weak Blackwell monotonicity without loss of generality.

The characterizations here exploit results and intuition from classical consumer theory. 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 can be viewed as the same object. In the consumer setting, expenditure is computed as the inner-product of price and quantity demanded. Similarly, the ex-ante payoff from the experiment can be computed as the inner-product of the information structure and posterior value function. Thus, the ex-ante payoff can be treated as wealth.

The similarity between standard consumer theory and costly information acquisition extends beyond the correspondence of primitives. The nonseparable model of costly information acquisition is defined as a preference that is increasing in ex-ante payoff and depends on the information structure. Similarly, in consumer theory, the indirect utility function is increasing in wealth and depends on prices. For costly information acquisition, the utility of a menu is obtained through maximization with respect to information structures; while in consumer theory, the utility of a consumption bundle can be obtained through minimization of the indirect utility over price vectors. While the optimization principle differs, the same underlying duality holds, which leads to the characterization by a condition resembling the General Axiom of Revealed Preference.

While we have highlighted the similarity to standard consumer theory, there are some technical differences. Most importantly, the information structures and posterior value functions are objects in infinite dimensional vector spaces. Thus, our proofs utilize the general results on quasi-concave duality that have been fruitfully studied by Chateauneuf and Faro (2009) and Cerreia-Vioglio et al., 2011a, Cerreia-Vioglio et al., 2011b. However, once one makes this connection the results follow by leveraging existing revealed preference and duality techniques. As such, the paper also serves as a didactic exercise.

This paper is related to other works on costly information acquisition and boundedly rational behavior. Costly information acquisition has received study from various perspectives in Denti et al. (2016), Ellis (2018), and Matejka and McKay (2014). Costly information acquisition has received study from a revealed preference perspective in Caplin and Martin (2015) and Caplin et al. (2015). Boundedly rational behavior has been studied with revealed preference conditions in Fudenberg et al. (2015), Aguiar (2016), and Allen and Rehbeck (2019).

The paper proceeds as follows. Section 2 presents the notation and some useful facts. Section 3.1 introduces and characterizes the nonseparable costly information acquisition model. Section 3.2 presents a model with a multiplicative cost of information acquisition. Section 3.3 presents a variant of the model whereby choice of information structure is costless, but is constrained to lie in some unknown set. Section 4 compares the nonseparable model to the additive model in Caplin et al. (2015), focusing on both the gap between behavioral axioms and out-of-sample predictions. Section 5 discusses some limitations on when violations of the conditions can be detected. Section 6 discusses methods to numerically deal with unknown utility numbers and prior distributions. Finally, Section 7 contains our concluding remarks. Proofs are relegated to the appendix.

Section snippets

Notation

We study a decision maker facing actions with state-contingent payoffs.4 Notation is consistent with Caplin et al. (2015) whenever possible for ease of comparison. We study a variety of models that are increasing in ex-ante payoff 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

Characterizing costly information models

In this section, we introduce three models of costly information acquisition. The nonseparable information cost model is the most general: it assumes only that the decision maker prefers higher ex-ante expected payoffs from choices and that more informative signals are more costly. Both the multiplicative cost model and the constrained information model are special cases of the nonseparable model.

Relationship between nonseparable and additive model

As a point of reference, we examine in detail how the nonseparable costly information representation relates to the additive costly information representation in Caplin et al. (2015).

We first review the definition of an additive costly information model, and show that the nonseparable model generalizes the additive model. We then show that one particular limitation of the additive model is that it forbids individuals from choosing less information whenever the menu provides a higher return to

Limitations

The revealed preference conditions for costly information acquisition often provide interesting bounds and intuition for these models. Moreover, we note that an additive costly information representation has the property of being translation invariant in ex-ante payoff. Similarly, a multiplicative costly information representation has the property of being scale invariant in ex-ante payoff.

One may want to look at choices from menus of this type to violate an additively separable or

Unknown utility, unknown prior

The model we examined requires utility and the prior to be known. In fact, we were able to write the revealed information structures in this “reduced-form” only because the prior probabilities are known. With that being said, even if the utility is unknown, then some implications of the model may be derived. As a general rule, if utility is totally unrestricted, then the model has no content. This is a relatively standard observation, and owes to the fact that complete indifference can

Conclusion

In this paper, we provide revealed preference characterizations for several models of costly information acquisition. The most general form allows for costs from time delay in addition to an additively separable cost. The characterization of these models follows directly from classical revealed preference theory. We also provide examples showing how the information acquisition differs across models.

References (35)

  • Stephen G. Bronars

    The power of nonparametric tests of preference maximization

    Econometrica

    (1987)
  • Donald J. Brown et al.

    The nonparametric approach to applied welfare analysis

    Econ. Theory

    (2007)
  • Andrew Caplin et al.

    A testable theory of imperfect perception

    Econ. J.

    (2015)
  • Andrew Caplin et al.

    Revealed preference, rational inattention, and costly information acquisition

    Am. Econ. Rev.

    (2015)
  • Simone Cerreia-Vioglio et al.

    Complete monotone quasiconcave duality

    Math. Oper. Res.

    (2011)
  • Christopher P. Chambers et al.

    Revealed Preference Theory, vol. 56

    (2016)
  • Christopher P. Chambers et al.

    Testing theories of financial decision making

    Proc. Natl. Acad. Sci.

    (2016)
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    We thank Yaron Azrieli, Mark Dean, and Pietro Ortoleva for useful comments and suggestions. We also thank an anonymous referee for correcting an error in an earlier draft of the paper.

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