Belief updating and the demand for information

Highlights

• Individuals differ consistently in responsiveness to information.

• The trait explains behavior across different choice environments.

• Valuation of useful information underreacts to increased informativeness.

• People value information that may yield certainty disproportionately highly.

• Biases mainly due to non-standard belief updating rather than risk preferences.

Abstract

How do individuals value noisy information that guides economic decisions? In our laboratory experiment, we find that individuals underreact to increasing the informativeness of a signal, thus undervalue high-quality information, and that they disproportionately prefer information that may yield certainty. Both biases appear to be mainly due to non-standard belief updating. We find that individuals differ consistently in their responsiveness to information – the extent that their beliefs move upon observing signals. Individual parameters of responsiveness to information have explanatory power in two distinct choice environments and are unrelated to proxies for mathematical aptitude.

Introduction

Modern economies increasingly trade not just in physical goods, but in information. Economic agents consider whether to purchase medical tests, consulting expertise, or financial forecasts. Entire industries are premised on the sale of useful but noisy information.

In this paper we use a laboratory experiment to investigate whether, and why, the demand for information systematically deviates from the predictions of the standard rational agent model. In contrast to the existing literature on the demand for information, information in our setting is instrumentally useful but concerns a state that is of no intrinsic relevance to subjects, and strategic considerations play no role.

The purchase of information is commonly conceptualized as the decision to partake in a two-stage lottery. In the first stage, the agent updates her beliefs upon observing the realization of an informative signal. In the second stage, she chooses an action, upon which the state of the world is revealed and payoffs are realized.¹ A standard rational agent will update beliefs according to Bayes' rule, and make choices according to expected utility theory. For any agent who correctly anticipates his own belief updating, this suggests two mechanisms through which empirical behavior could deviate from the predictions of the standard model.

First, the demand for information may be affected by systematic deviations from expected utility theory in compound lotteries. These have been documented and modeled, for instance, by Bernasconi and Loomes (1992), Ergin and Gul (2009), Halevy (2007), Halevy and Feltkamp (2005), Segal (1987), Seo (2009), and Yates and Zukowski (1976). Second, the demand for information may be influenced by biases in belief updating (reviewed in Camerer, 1995). Our design allows us to estimate the effect of the latter channel on the demand for information, and to identify three biases in belief updating and the demand for information.²

In our experiment, subjects reveal how valuable they think it is to first observe an informative signal about a binary state of the world and then make a guess about it. At the very end of the experiment, a correct guess may be rewarded with a prize. Since our elicitation is in terms of probability units of winning the prize, our results are independent of the shape of subjects' utility function for money. Discriminating between channels is possible as a second stage asks subjects to reveal posterior beliefs for each of these signals. A third stage presents subjects with a conditionally i.i.d. sequence of signal realizations that depends on the state of the world. It measures how much information subjects require until they first prefer betting on the state of the world to an outside option.

We identify two biases in the demand for information that depend on the properties of information structures. Subjects value signals of varying informativeness as though they were more alike, compared to the predictions of the standard model (the compression effect). Additionally, subjects disproportionately prefer information structures that may perfectly reveal the state of the world (the certainty effect). We also test how valuations change if we increase the precision of the signal in one state of the world and decreasing it in another, such that the utility of a standard rational agent is unchanged. A class of theories of non-standard risk preferences predict that this will make an information structure less attractive to subjects (Segal, 1987; Seo, 2009). Our data do not support this hypothesis.

We identify a third bias that is individual-specific. Subjects in our experiment are heterogeneous in responsiveness to information, the extent to which their beliefs move upon observing information. Importantly, this heterogeneity is consistent within individuals. We measure this individual tendency in one task, and find that it is significantly correlated with subjects' behavior in two additional tasks. Moreover, while it is related to cognitive style, it is uncorrelated with proxies for mathematical aptitude. In particular, neither the direction nor the absolute size of this bias is correlated with whether a subject knows Bayes' law, has taken a statistics class, or has a STEM major.

Following Grether (1980) and Holt and Smith (2009), we propose a one-parameter model of responsiveness to information and estimate each subjects' parameter using data only from the latter part of our experiment. This procedure collapses all the data about each subject's belief updating to a single parameter, yet it explains 80% of the individual variation in the demand for information that can be explained with a model that uses each individual's beliefs data in a maximally flexible way. Our individual-level estimates of responsiveness to information are also significantly related to subjects' behavior in our environment that gradually uncovers information and reveals how quickly subjects' beliefs about the state of the world cross a given threshold.³

Conditional on the estimated parameters, the responsiveness model also generates the compression effect and the certainty effect. This is perhaps surprising, since the model is set up to capture a single individual-specific bias, while these two effects pertain to properties of the information structures. Both of these are consequences of the fact that our average subject is less responsive to information than a Bayesian, but correctly interprets perfectly informative and perfectly uninformative signal realizations. Formally, the model of responsiveness to information is equivalent to applying a prospect theory probability-weighting function to the Bayesian posterior. The information structure-specific effects we observe are implications of the inverse S-shape of that function.

Many of our results could in principle be due to either non-Bayesian belief updating, or due to non-standard risk preferences. We argue that the latter are unlikely to explain the entirety of our data, for three reasons. First, by design, our results cannot be due to the shape of subjects' utility function for money. Each choice in our experiment elicits, for some event X, the value of k such that the subject is indifferent between receiving $35 with probability k and $0 otherwise, and receiving $35 if event X occurs and $0 otherwise.⁴ For any expected utility maximizer, the value of k that leaves such an agent indifferent does not depend on his degree of risk aversion (Roth and Malouf, 1979).

Second, also by design, they cannot be due ambiguity aversion, at least if interpreted in the standard fashion. In our experiment, all probabilities are objectively given. Subjects have induced priors about the state space, and all information structures are explicitly given, so that the probability of any payoff-relevant event can be calculated, via a simple application of Bayes' rule. In contrast, ambiguity aversion models are usually interpreted to apply under Knightian uncertainty, where objective probabilities are not applicable (Knight, 1921; Ellsberg, 1961; Baillon and Bleichrodt, 2015).

Third, there are other accounts of decision making under risk that are not excluded by our design choices. These do not account for the compression effect in the valuation of information structures, however, as this effect vanishes once we account for elicited belief updating data. The extent of the boundary effect is affected by controlling for beliefs data, but some of it remains even then. This suggests that it is partially rooted in risk preferences.⁵

Our study contributes to the literature in several ways. While responsiveness to information relates to both base-rate neglect (Kahneman and Tversky, 1972) and conservatism (see Peterson and Beach, 1967 for a review), we show that it is heterogeneous across subjects, stable within subjects, and has explanatory power across three choice environments.⁶ We explicitly relate biases in belief updating to prospect theory probability weighting (as reviewed, for instance, in Fehr-Duda and Epper, 2012).

A handful of papers have studied individual heterogeneity in belief updating. Most closely related is the successor to our paper by Buser et al. (2016) who also find, in a setting with ego-relevant beliefs, that responsiveness to feedback is a personal trait that predicts selection into competitive environments. (They do not relate the trait to information demand, and do not consider variation in information structures.) Augenblick and Rabin (2015) show that if a forecaster's belief stream for a given geopolitical event is relatively volatile, then the same forecaster's belief stream for other events is also relatively volatile. El-Gamal and Grether (1995) fit a finite mixture model to binary-choice data, and find that the most frequently used updating rules are Bayes, base-rate neglect (which roughly corresponds to excessive responsiveness), and conservatism (which is equivalent to insufficient responsiveness). These authors do not study individual consistency across different kinds of updating tasks.⁷ Our paper complements Antoniou et al. (2017) who document individual heterogeneity in how strongly individuals react to the salience of information, keeping constant its diagnosticity.

Within the empirical literature on the demand for information, our study contributes to the literature by investigating biases pertaining to different information structures (characterized by their state-dependent signal accuracies), and by relating them to a stable individual trait that characterizes belief updating. These results distinguish it from Hoffman (2016) who studies internet-domain-name traders in a lab-in-the-field experiment. He finds that the willingness to pay (WTP) for information increases with a subject's uncertainty about the state of the world, increases with the accuracy of the signal, as predicted by the Bayesian benchmark, but does so to a lesser extent than predicted by theory. This is related to the compression effect we document in the demand for information.

A related literature documents that subjects do not always sufficiently account for the properties of the data generating process when making inferences (Massey and Wu, 2005; Fiedler and Juslin, 2006). Our finding that subjects' are willing to pay too much for low-quality information, and not willing to pay enough for high-quality information complements Falk and Zimmermann (2016) who find that in a context of non-instrumental information, subjects prefer few realizations of high-quality information to many realizations of low-quality information.⁸

We know of no further empirical work on the demand for information in a non-strategic setting in which information is instrumentally useful but concerns a state that is of no intrinsic relevance to subjects, and strategic considerations play no role. Outside of our domain, Eliaz and Schotter (2010) show that subjects are willing to pay for information that will not change their decision, if it decreases their uncertainty about having taken the right action, and Bastardi and Shafir (1998) find that once subjects decide to purchase information, they base their decisions on it, even if they would not do so if they were given the information for free. Masatlioglu et al. (2015) study how the demand for non-instrumental information varies with the properties of the available information structures. Burks et al. (2013), Eil and Rao (2011), and Moebius et al. (2013) find that when beliefs are ego-relevant, WTP for information is larger the more confident the subject is that information will be desirable. Kocher et al. (2014), Falk and Zimmermann (2016), and Zimmermann (2014) study preferences over the dynamic arrival of information. The first of these papers finds that subjects prefer later resolution of uncertainty about a lottery ticket, the second finds that they prefer later resolution about whether an unpleasant event will occur if they are not forced to focus on that event, and the third that subjects display no preference over clumped vs. piecewise information regarding a lottery ticket. The demand for dynamic information is also studied in the literature on sequential search (for instance Brown et al., 2011; Caplin et al., 2011, and Gabaix et al., 2006) and in the literature on explore–exploit dilemmas (see Erev and Haruvy, 2015 for a recent review). Finally, endogenous information acquisition has been studied in a wide range of strategic settings. For example, Kubler and Weizsacker (2004) focus on social learning and informational cascades, Szkup and Trevino (2015) on global games, Gretschko and Rajko (2015) on auctions, and Sunder (1992) and Copeland and Friedman (1987) on experimental asset markets.

The remainder of this paper proceeds as follows: Section 2 presents the experimental design and our hypotheses. Section 3 analyzes behavior as a function of the information structure, documents the compression and certainty effects, and argues that they are likely due to belief updating biases rather than due to non-standard risk preferences. Section 4 studies individual-level consistency, presents our results regarding responsiveness to information, and links them to the information-structure specific biases. Finally, section 5 discusses potential applications of our findings and concludes.