Rational Inattention and Choice under Risk: Explaining Violations of Expected Utility through a Shannon Entropy Formulation of the Costs of Rationality

Article excerpt

Introduction

The falsification of expected utility theory (EU) through experiment has led to a fruitful literature over the past 50 years in order to try to account for this discrepancy. While at its beginnings and through the 1980s, this field was primarily concerned with constructing deterministic alternatives to expected utility theory (Allais 1953, Kahenman and Tversky 1979, Loomes and Sugden 1982), following the seminal work of Hey and Orme (1994), researchers in this area began to consider choice functionals that were stochastic, cither with preferences themselves being random, or choices being distorted from optimal through some kind of random noise. This has been a very fruitful development, as the explicit modeling of error in choices has allowed the standard tools of econometrics to be used in testing hypotheses concerning whether observed experimental patterns are statistically distinguishable from those predicted by expected utility theory. However, the obvious shortcoming of this approach has been the introduction of potentially erroneous assumptions through misspecification of the error term, as its form is usually taken as axiomatic rather than being derived from some maximization problem. Remedying this problem would put stochastic choice theory on a solid theoretical footing and, hopefully, improve our explanation of violation of expected utility in experiments.

We propose a model of stochastic choice in which the error term is derived from a maximizing framework in which information is costly for agents. Following Sims (2003, 2006), we assume that agents' rational faculties, like physical communication channels, are capacity constrained: they can transmit only a finite amount of information in a finite amount of time, and therefore, agents must choose probabilistically. As in Sims's model, we define agents to be rationally inattentive in the sense that when constrained to choose randomly out of the feasible set, agents optimally select the distribution of the error term of their decisions. We follow Sims in representing the information constraint that individuals face as a constraint on the degree of dependence between agents' actions and the data of the choice problem that they observe. When the agents' actions and the data of their choice problem are viewed as random variables, their degree of dependence is their mutual information computed using Shannon entropy. (1)

We show that the model is a special case of a discrete choice entropy model solved in Woodford (2007), and arrive at an explicit formula for the probability of choosing either lottery conditional on the data of the lottery choice problem. Finally, we demonstrate that under homogeneity assumptions, the model has testable implications and can be taken to data in a straightforward manner, as it is equivalent to the logit model of binary choice.

Estimating the model over laboratory data from Loomes et al. (2002), Hey (2001) and Hey and Orme (1994) we confirm that the error term specification is superior to the white noise (Gaussian) error term that is currently used in the literature. This finding is important as it provides evidence that errors are products of rational inattention rather than essentially random factors. We use the Shannon entropy error specification to test expected utility theory against alternatives in the literature, and discover that 1) EU is significantly dominated by at least one alternative model for the vast majority of individuals, and 2) the rank-dependent models perform noticeably better than all other alternative models, but 3) no single alternative model is significantly superior to EU for a significant to overwhelming fraction of individuals. We conclude by considering the goodness-of-fit measures of deterministic and stochastic models of choice under risk.

Literature Review

Expected Utility Theory and Decision Under Risk

The literature on expected utility theory and its alternatives is too voluminous to be given full justice here; a good review is Starmer (2000). …