Abstract
In his exposition of subjective expected utility theory, Savage (1954) proposed that the Allais paradox could be reduced if it were recast into a format which made the appeal of the independence axiom of expected utility theory more transparent. Recent studies consistently find support for this prediction. We consider a salience-based choice model which explains this frame-dependence of the Allais paradox. We then derive the novel prediction that the presentation format responsible for reductions in Allais-style violations of expected utility theory will also reduce Ellsberg-style violations of subjective expected utility theory. This format makes the appeal of Savage’s “sure thing principle” more transparent. We design an experiment to test this prediction and find strong support for such frame-dependence of ambiguity aversion in Ellsberg-style choices. In particular, we observe markedly less ambiguity-averse behavior in Savage’s matrix format than in a more standard “prospect” format. This finding poses a new challenge for the leading models of ambiguity aversion.
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Notes
We are speaking here of the classic Allais example, which is a thought experiment involving very large hypothetical outcomes which no experimenter can actually pay out. When its outcomes are proportionally scaled down to an experimentally feasible size for actual payment, similar behavior does not always occur (e.g. Conlisk 1989; Fan 2002). This could be either a payoff magnitude effect or a hypothetical versus real incentives effect. In incentivized experiments, the generalized Allais example (known as the common consequence effect) does not always occur (Burke et al. 1996) and sometimes occurs in “non-classic” ways (Starmer 1992).
A minimal frame is a matrix presentation of choice alternatives which (among other properties) has the smallest dimension (fewest number of columns) needed to represent those alternatives. See Leland and Schneider (2016) for formal property lists which uniquely define minimal and transparent frames.
Another SWUP variant would add a probability ϑ ∈ [0, 1] that the agent naturally re-frames transparent presentations as minimal ones. This is particularly plausible if people naturally think in minimal frames. ϑ then governs the strength of the framing effect for that agent (agents with ϑ = 1 are frame insensitive and agents with ϑ = 0 conform to SEU in transparent frames but exhibit ambiguity aversion in minimal frames). This SWUP variant accommodates reduced ambiguity aversion (without requiring ambiguity neutrality) in transparent frames, but still rules out ambiguity seeking behavior. Hurwicz-SWUP allows any ambiguity attitude, but requires less ambiguity aversion (or more ambiguity seeking) in transparent frames than in minimal frames.
While there is not yet a general consensus for ranking all pairs of ambiguous acts by their level of ambiguity, one natural approach is given by the family of f-divergences which measures the distance between two probability distributions. Online Appendix Section A3 shows that two well-known f-divergences—the Hellinger distance (Hellinger 1909; Sengar 2009) and the total variation distance (Levin et al. 2009) predict the same ranking of ambiguous acts for each of the basic pairs in our experimental design.
Consider a hypothetical noiseless subjective EV maximizer Bob with equal priors over ticket colors. His expected probability of receiving $25 in our design would have been 0.518056 (and otherwise zero). Now assume a sample of 79 such Bobs. Simulation of a million such samples show that average earnings of those 79 Bobs will exceed $11.01 in 93% of those samples. Now consider a hypothetical random chooser Ted. His probability of receiving $25 would be 0.492361 in our design, and the average earnings of 79 Teds will exceed $11.01 in 84% of samples.
The planned sample was 80 subjects. One subject failed to show for the final session.
Each of the five experimental sessions lasted approximately 90 minutes (in keeping with recruitment promising that sessions would be less than 2 hours).
For interpretation of results and estimation we assume that any prior probabilities subjects place on the red and blue ticket states are constant across their choice situations. Our placement of the bags with the subjects, from the start to the finish of their session, is meant to make this assumption plausible.
For instance, in Option B in Basic Pair 1, any die roll between 1 and 9 paid $25 and any die roll between 10 and 12 paid $0 if a red ticket was drawn. Likewise, any die roll between 1 and 3 paid $25 and any die roll between 4 and 12 paid $0 if a blue ticket was drawn.
In the econometrics of risk and uncertainty, structural estimation at the individual subject level benefits strongly from choice problems which challenge the boundaries of each subject’s attitudes toward risk and/or ambiguity. Thus the inclusion of a range of choice problems, some that tempt even relatively ambiguity-averse subjects to choose the ambiguous option (e.g. Option B in basic pair 6) add significant information concerning preferences. Minimal frames predominate in our design. This enables the “generalization criterion” analysis we perform below in Section 5.3. Additionally, note that transparent frames provide no information about probability salience functions ψΡ. In other work based on these data, we plan to test properties of these particular salience functions.
Computation of ambiguity premia requires knowledge of von Neumann and Morgenstern (vNM) functions, but below we find that some interesting features of ambiguity premia are fairly robust to curvature of vNM functions.
MSD preferences include many other preferences. Grant and Polak show that since the standard deviation dispersion function is non-negative, convex, symmetric, and satisfies certainty betweenness, MSD preferences have a corresponding representation in the vector expected utility model (Siniscalchi 2009), the invariant biseparable representation (Ghirardato et al. 2004), and the multiple prior representation (Gilboa and Schmeidler 1989) provided that the mean-standard deviation preferences are monotone (a property we impose on our estimations of MSD preferences, though this monotonicity constraint rarely binds at the level of individual subjects).
By “incentive-compatible” we mean that the subject’s choice in any one of the several tasks will be equivalent to the choice the subject would have made in that task if that task had been the sole task presented to the subject.
For every sign test result reported in this article, we also calculated results of the Wilcoxon signed rank test and a paired sample t-test. Those two tests always yield still smaller p-values than the sign tests do.
Let SEV(X) denote the subjective expected value of option X under equal priors πr = πb = 0.5. Define the total premium SEV(X) − CE(X| α) and the risk premium SEV(X) − CE(X| 0.5). Our definition of the ambiguity premium simply decomposes the total premium into the sum of the risk premium and ambiguity premium.
One sometimes thinks of \( \Delta {k}^{\tau_1,{\tau}_2} \) as a penalty for relative lack of parsimony. More accurately, it is due to a difference in degrees of freedom lost because the very observations used to compute fit were also used to estimate parameters. This explains why the generalization criterion has no such “penalty:” Observations used to calculate fit (those in \( \mathcal{T} \)) were not used to estimate θτ (that estimation used only observations in \( \mathcal{M} \)).
This long analysis cannot be include here (the third author will provide it on request). Its essence is to allow both parameters governing attitude toward ambiguity, and probabilistic model parameters governing the noisiness of decisions, to depend on whether a situation is in a minimal or transparent frame. We did this using the Mean-Standard Deviation theory, both with and without random parameter controls for heterogeneity across subjects. No specification uncovers significant increases in the noisiness of our subjects’ decisions in transparent frames and, in all of those specifications, we find a significant decrease in their ambiguity aversion in transparent frames.
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Acknowledgements
We thank an anonymous referee and Glenn Harrison for helpful comments, as well as seminar participants at Chapman University, the University of Michigan, the 2016 Edwards Bayesian Research Conference, and the 2016 Bay Area Behavioral and Experimental Economics Workshop. The views expressed are those of the authors and do not necessarily reflect those of the National Science Foundation or the United States.
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Schneider, M., Leland, J.W. & Wilcox, N.T. Ambiguity framed. J Risk Uncertain 57, 133–151 (2018). https://doi.org/10.1007/s11166-018-9290-6
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DOI: https://doi.org/10.1007/s11166-018-9290-6
Keywords
- Ellsberg paradox
- Ambiguity aversion
- Framing effects
- Expected utility
- Allais paradox
- Salience theory
- Independence axiom