[Paper Review] Approximate Expected Utility Rationalization

Federico Echenique, Kota Saito+1 more|arXiv (Cornell University)|Feb 12, 2021

Decision-Making and Behavioral Economics4 citations

TL;DR

This paper introduces a novel measure of deviation from expected utility (EU) theory by quantifying how close a dataset is to being rationalized by EU, using a parameter $e$ that bounds discrepancies in beliefs, utilities, or perceived prices. The method detects violations of key EU properties—such as downward-sloping demand and stochastic dominance—more sensitively than the Critical Cost Efficiency Index (CCEI), and empirical applications on large-scale risk-choice experiments reveal significant gaps between general utility maximization and EU consistency.

ABSTRACT

We propose a new measure of deviations from expected utility theory. For any positive number~$e$, we give a characterization of the datasets with a rationalization that is within~$e$ (in beliefs, utility, or perceived prices) of expected utility theory. The number~$e$ can then be used as a measure of how far the data is to expected utility theory. We apply our methodology to data from three large-scale experiments. Many subjects in those experiments are consistent with utility maximization, but not with expected utility maximization. Our measure of distance to expected utility is correlated with subjects' demographic characteristics.

Motivation & Objective

• To address the limitation of revealed preference methods that treat consistency with expected utility as binary, by introducing a continuous measure of deviation from EU theory.
• To develop a method that captures violations of fundamental EU properties—such as downward-sloping demand and first-order stochastic dominance—that are ignored by traditional measures like CCEI.
• To provide a theoretically grounded and empirically testable framework for assessing how close observed choice data is to rationalization under expected utility, using a parameter $e$.
• To demonstrate the method’s superiority over CCEI through theoretical analysis and empirical validation on large-scale experimental datasets.
• To explore the correlation between distance to EU and demographic characteristics, offering new insights into behavioral heterogeneity in risk preferences.

Proposed method

• Proposes an $e$-approximate rationalization framework where a dataset is considered within $e$ of expected utility if there exists a utility function, belief structure, and perceived prices such that all observed choices are within $e$ of EU-optimality.
• Introduces the concept of $e$-PSAROEU (approximate rationalization with bounded error) to formalize the notion of how close a dataset is to satisfying EU axioms.
• Uses a revealed preference axiomatization based on observed prices and consumption to characterize datasets that are $e$-close to EU rationalization.
• Employs a perturbation argument using rational approximations of real numbers (e.g., log prices, probabilities, and $e$) to ensure the existence of rational solutions close to real-valued ones.
• Applies a duality-based proof strategy using linear systems and separating hyperplanes to show that if no $e$-rationalization exists, then a contradiction arises under certain conditions.
• Leverages rational approximations of irrational numbers (e.g., $\log(1+e) \in \mathbb{Q}$) to construct a sequence of rationalized datasets that converge to the original in terms of $e$-closeness and probability measures.

Experimental results

Research questions

• RQ1 How can we measure the degree to which observed choice data deviates from expected utility theory, rather than treating consistency as a binary outcome?
• RQ2 Why is the Critical Cost Efficiency Index (CCEI) inadequate for detecting violations of key EU properties such as downward-sloping demand and stochastic dominance?
• RQ3 Can a continuous measure of distance to EU rationalization be constructed that is both theoretically sound and empirically useful?
• RQ4 How do demographic characteristics correlate with the degree of consistency with expected utility theory in experimental data?
• RQ5 To what extent do subjects who are consistent with general utility maximization (by CCEI) still violate expected utility axioms, and how can this be quantified?

Key findings

• The proposed $e$-measure detects violations of downward-sloping demand and first-order stochastic dominance—key properties of EU—more effectively than CCEI, which fails these 'smell tests'.
• In three large-scale experimental datasets (Choi et al., 2014; Carvalho et al., 2016; Carvalho and Silverman, 2019), many subjects who are consistent with general utility maximization (CCEI ≈ 1) still exhibit significant deviations from expected utility, as measured by the new $e$-metric.
• The distance to EU is correlated with demographic characteristics, and these correlations are intuitive and consistent with behavioral economics predictions.
• The method successfully rationalizes datasets that are not rationalizable under standard EU, by allowing bounded deviations in beliefs, utilities, or perceived prices.
• Theoretical analysis confirms that the $e$-measure is more sensitive to structural violations of EU than CCEI, particularly in cases involving stochastically dominated choices.
• The existence of rational approximations (e.g., rational log prices and probabilities) ensures that the $e$-rationalization problem can be solved in a computationally feasible way, even when real numbers are involved.