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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). The pioneering articles to model randomness in choice under risk explicitly were Harless and Camerer (1994), who assumed that individuals had a fLxed probability in each lottery choice problem of choosing each lottery with equal probability, and Hey and Orme (1994), who supposed that the utility difference between the two lotteries (or the binary choice function) was perturbed by a normal white noise (Gaussian) (2) error term. After these two articles, and after a new wave of experiments that were more congenial to an econometric analysis of the data, the literature on EU shifted towards stochastic specifications. Loomes, Moffat and Sugden (2002) compare different ways of modeling the stochastic aspect of preferences, as well as allow the stochastic component of choice to change as the experiment progresses. Using specifications derived from a search, they reject expected utility theory, but argue that behavior during the experiment converges to that predicted by expected utility theory in the long run. Hey (1995) and Buschena and Zilberman (2000) experiment with making the error term heteroskedastic by allowing it to depend on the number of states in the lotteries or on the amount of time spent on a choice problem. They report that under heteroskedastic specifications, a much larger percentage of subjects behave consistently with expected utility theory. These developments suggest that deviations from expected utility may be explained by the stochastic nature of choice rather than by an explicit violation of the core axioms of expected utility by the subjects' underlying preferences.
Shannon Entropy and Decision Costs
The entire subject of information theory rests on the concept of Shannon entropy; the pioneering article is Shannon (1948), a classic textbook is Cover and Thomas (1991), anda source for an axiomatic analysis of Shannon entropy is Csiszar and Korner (1981). Shannon entropy has been used in economics as an estimation criterion in econometrics, and extensively in finance within the analysis of prediction of asset behavior, but Sims (1998) is one of the first articles to introduce it as a measure of rational inattention. Sims (2003) considers its implications in macroeconomics, while Sims (2006) solves the general case of an elementary consumption-savings problem under the assumption of fully divisible income, and shows that these choices can have a discrete character without any inherent lumpiness in the deterministic process or in the marginal distribution of wealth. Woodford (2007) is an application of Sims's idea to the field of price-changing behavior. Work on generalizing Sims (2006) to a wider intertemporal choice setting is being done by Luo (2006).
Theoretical Formulation of the Optimization Problem and its Solution
Suppose that the data of a decision problem are given by the vector [xi], and the individual's sole decision variable is a binary variable a. Define P([xi]) := P(a = 1|[xi]) as the probability the individual sets a to 1 given data [xi]. Then, the expected utility of the individual for given data [xi] and decision function P is [E.sub.p](U(A,[xi]))= P([xi])U(1,[xi]) + (1 -P([xi]))U(0,[xi]) = P([xi])L([xi]) + U(0,[xi]) where we denote L([xi]):= U(1,[xi]) - U(0, [xi]) as the gains from an optimal decision, or the loss from a suboptimal decision. If it were effortless to think, and there were no costs of rationality, P would be trivially given by P([xi]) = 1 [L([xi])[greater than or equal to] 0]. However, we instead assume that thinking is costly. Therefore, instead of there being a mapping from input variables [xi] into actions a, there may be a stochastic component in the relationship, as though [xi] determined a only imperfectly. While from the point of view of the (rationally unconstrained) observer, we may think that a is a random variable (hereafter A) whose realization is conditional on [xi], from the point of view of the agent's reason, we may think of (hereafter X) as a random variable, since it is impossible to map it perfectly into action. The agent can then be assumed to have a prior distribution g([xi]) in mind over this random variable X, and the agent's problem becomes to choose an optimal (possibly random) course of action when confronted with a realization of [xi] from g, while taking into account the costs of rationality.
As we discussed in the introduction, these costs may be modeled by the amount of information obtained about what the value of A should be from observing the realization of the variable X, which is
H(A) + H(X) - H(A,X)
where H is Shannon entropy. This is the difference between the Shannon entropy of the random vector (A,X) and the entropy of this vector if A and X were independent, which is the sum of the entropies of the components (Cover and Thomas 1991). Hence, the agent's problem becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where the LHS of the constraint is the mutual information between A and X and k is a bound on the amount of information that can be acquired. As is shown in Woodford (2007), the constraint can be rewritten in terms of P([xi]) and g([xi]), so that the optimization problem becomes:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
This is a problem in the calculus of variations, and is solved in Woodford (2007). The optimal decision function is:
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The parameter [lambda] is the Lagrange multiplier of the Shannon entropy constraint, and corresponds to the utility cost of information.
This formula is a crucial result both for theoretical consideration of the problem and for its empirical testing. We immediately see that the agent acts as an expected utility maximizer ([P.sub.[lambda]](a = 1|[xi]) = I[L([xi]) > 0]) if and only if [lambda]=0, which is equivalent to saying that information is free. Therefore, we see already that introducing Shannon entropy constraints induces choice behavior that violates expected utility theory, despite the fact that the underlying utility of the agent is of the expected utility form. In contrast to much of the existing literature on decision theory, this result derives expected utility violations from an explicit maximizing framework with a very natural assumption about the costliness of attention to information and a highly plausible functional form for this cost.
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