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Decision theory and the "almost implies near" phenomenon

Christopher P Chambers, Federico Echenique

TL;DR

The paper develops a unified, axiom-centered framework showing that when core decision-theoretic axioms are satisfied approximately (through observable violations like $oldsymbol{ extvarepsilon}$ or $oldsymbol{ extTheta}$), there exists a pair of utility functions—one representing observed behavior and one satisfying the exact standard axiom (e.g., expected utility or exponential discounting)—that are closely aligned. It provides quantitative links between the size of axiom violations and the distance between the corresponding representations, across risk, uncertainty, and intertemporal choice, via the EU, Anscombe–Aumann, and dated-rewards models. The results imply that small deviations from rationality yield near-rational representations, thereby justifying the use of standard models as accurate approximations even when axioms are not perfectly satisfied. The work also clarifies when approximate representations can be exact (e.g., under homothety or specific stationarity conditions) and discusses implications for models like MEU and smooth ambiguity, highlighting the practical relevance for empirical work and approximate optimization.

Abstract

We examine behavioral axioms in decision theory that are satisfied approximately rather than exactly. We demonstrate that in key domains -- decisions under risk, uncertainty, and intertemporal choice -- behavior that \emph{almost} satisfies an axiom implies the existence of a utility function that is \emph{near} one that adheres to the standard theoretical representation (e.g., expected utility, or exponentially discounted utility). We explicitly quantify the distance between the utility that captures actual behavior and the ideal theoretical utility as a function of the measured deviation from the axiom. This result formally connects two distinct quantitative exercises: measuring empirical deviations from theory and utilizing approximate optimization. Effectively, we show that small deviations from behavioral axioms rationalize the use of standard models as valid approximations.

Decision theory and the "almost implies near" phenomenon

TL;DR

The paper develops a unified, axiom-centered framework showing that when core decision-theoretic axioms are satisfied approximately (through observable violations like or ), there exists a pair of utility functions—one representing observed behavior and one satisfying the exact standard axiom (e.g., expected utility or exponential discounting)—that are closely aligned. It provides quantitative links between the size of axiom violations and the distance between the corresponding representations, across risk, uncertainty, and intertemporal choice, via the EU, Anscombe–Aumann, and dated-rewards models. The results imply that small deviations from rationality yield near-rational representations, thereby justifying the use of standard models as accurate approximations even when axioms are not perfectly satisfied. The work also clarifies when approximate representations can be exact (e.g., under homothety or specific stationarity conditions) and discusses implications for models like MEU and smooth ambiguity, highlighting the practical relevance for empirical work and approximate optimization.

Abstract

We examine behavioral axioms in decision theory that are satisfied approximately rather than exactly. We demonstrate that in key domains -- decisions under risk, uncertainty, and intertemporal choice -- behavior that \emph{almost} satisfies an axiom implies the existence of a utility function that is \emph{near} one that adheres to the standard theoretical representation (e.g., expected utility, or exponentially discounted utility). We explicitly quantify the distance between the utility that captures actual behavior and the ideal theoretical utility as a function of the measured deviation from the axiom. This result formally connects two distinct quantitative exercises: measuring empirical deviations from theory and utilizing approximate optimization. Effectively, we show that small deviations from behavioral axioms rationalize the use of standard models as valid approximations.

Paper Structure

This paper contains 22 sections, 14 theorems, 93 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related literature.
  3. Motivation.
  4. Model and results
  5. Basic definitions and notation
  6. Expected Utility Theory
  7. Choice under uncertainty: the Anscombe-Aumann model
  8. Dated rewards
  9. Discrete time
  10. Strictly decreasing discount factors
  11. Continuous time
  12. Conclusion
  13. Proofs
  14. Proof of Theorem \ref{['thm:EU']}
  15. Proof of Theorem \ref{['thm:EU2']}
  16. ...and 7 more sections

Key Result

Theorem 1

If $\succeq$ satisfies the axioms, then there is a pair of functions $(u,\ell):\Delta\to [0,1]^2$ such that Conversely, if the pair $(u,\ell)$ satisfies these properties with $\| u-\ell \|_{\infty}<\varepsilon$, then $\succeq$ satisfies $4\varepsilon$--Reduction of compound lotteries.

Figures (1)

  • Figure 1: Utility differences: Cumulative Prospect Theory and Expected Utility.

Theorems & Definitions (20)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark
  • Corollary 4
  • Theorem 5
  • Theorem 6
  • Corollary 7
  • Theorem 8
  • Corollary 9
  • ...and 10 more