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How much ambiguity aversion?

Finding indifferences between Ellsberg’s risky and ambiguous bets

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Abstract

Experimental results on the Ellsberg paradox typically reveal behavior that is commonly interpreted as ambiguity aversion. The experiments reported in the current paper find the objective probabilities for drawing a red ball that make subjects indifferent between various risky and uncertain Ellsberg bets. They allow us to examine the predictive power of alternative principles of choice under uncertainty, including the objective maximin and Hurwicz criteria, the sure-thing principle, and the principle of insufficient reason. Contrary to our expectations, the principle of insufficient reason performed substantially better than rival theories in our experiment, with ambiguity aversion appearing only as a secondary phenomenon.

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Notes

  1. See the overview of the evidence presented by Camerer and Weber (1992). See also Fox and Tversky (1995), Keren and Gerritsen (1999), and Liu and Colman (2009).

  2. Ahn et al. (2010), Chow and Sarin (2001), Curley and Yates (1989), Hogarth and Einhorn (1990), Etner et al. (2012), Fox and Tversky (1995), Halevy (2007), Hey et al. (2010), Hsu et al. (2005), Trautman et al. (2008), Wakker (2010), and Abdellaoui et al. (2011). The link http://aversion-to-ambiguity.behaviouralfinance.net/ leads to many more papers on ambiguity aversion published since the year 2000.

  3. In manipulating the number of red balls to determine the extent of ambiguity aversion, our approach resembles that of MacCrimmon and Larsson (1979), Kahn and Sarin (1988), Viscusi and Magat (1992), and Viscusi (1997), each of which finds substantial ambiguity aversion. Our approach differs from the latter three studies in that rather than asking individuals outright how much the known probability would need to be to render them indifferent between a risky and uncertain bet, we use their choices between bets to estimate an indifference interval.

  4. The objective maximin criterion is often referred to as the minimax criterion. The confusion between maximin and minimax presumably arises because minimax equals maximin in Von Neumann’s famous minimax theorem. The confusion is sometimes compounded because Savage (1954) proposed a further decision criterion called the minimax regret criterion, which happens to make the same predictions as the maximin criterion in the special case considered in this paper. (Savage (1954, p. 16) distinguished between large and small worlds, recommending his minimax regret criterion for the former. He variously describes using Bayesian decision theory outside a small world as “preposterous” and “utterly ridiculous”.)

  5. Binmore’s (2009, p. 166) multiplicative version of the Hurwicz criterion also yields Eq. 3 to a first order of approximation. To a second order of approximation, it yields

    $$(3r_{1}-2)(3r_{2}-2)=1+c(r_{1}-r_{2})\{2(1-r_{1})(1-r_{2})- 1\}\,, $$

    for some small positive constant c.

  6. Psychologists favor the use of a titration over simply asking subjects for their indifference probabilities, but there is a risk that subjects might not always answer the titration questions truthfully because they prefer being paid on Ellsberg bets that can only be reached by lying. The monetary payoffs associated with each bet were chosen to make such misrepresentation unprofitable for risk-neutral subjects who honor the principle of insufficient reason, but ambiguity-averse subjects could sometimes gain by lying. However, it would be necessary for them first to learn what future questions would be asked before they could exploit the opportunity for misrepresentation, and we found no significant evidence of learning in comparing subjects’ behavior in later stages of the experiment.

  7. In this exercise, it is assumed that \(K\succ J\) when \(R=0\), and \(J\succeq K\) when \(R\geq \frac 12\); also \(L\succ M\) when \(R=0\), and \(M\succeq L\) when \(R\geq \frac 12\).

  8. For example, a subject who reveals values for \(r_{1}\) and \(r_{2}\) that both lie in the interval \([\frac {1}{3},\frac {3}{8}]\) (corresponding to the choices \(KJJ\) and \(LMM\)) will be regarded as satisfying the sure-thing principle (stp or STP). But our lax criterion also includes in STP a subject whose value of \(r_{1}\) is the same, but whose value of \(r_{2}\) lies in the neighboring interval \([\frac 27,\frac {1}{3}]\) corresponding to the choice \(MLL\) that might result if the subject made a misjudgment at the first question but answered later questions accurately.

  9. Savage (1954, p. 16) would perhaps have commented that leaving room for suspicion of dishonest manipulation by the experimenter creates a large world for the subjects. Our design is intended to make the subjects’ world small in this respect.

  10. We used Amazon’s Mechanical Turk (https://www.mturk.com), Psychological Research on the Net (http://psych.hanover.edu/Research/exponnet.html), and the Research Subject Volunteer Program (http://alkami.org http://alkami.org/). Participants recruited via Mechanical Turk were paid a token $0.05 for completing the approximately six-minute study.

  11. The prizes were approximately equal to their previous dollar values but were denominated in British pounds. The average payout was around \(\pounds \)13.

  12. In computing critical values of \(T\) for the 1%, 5%, and 10% levels, it is necessary to take account of the sample sizes, which vary between different versions of the experiment.

  13. The significance levels \(p\) have been computed from the K–S statistics using the formula \(p=k\times \{(n_1+n_2)/n_1n_2\}^{1/2}\), where \(n_{1}\) and \(n_{2}\) are the number of subjects in a population, and \(k=1.22\) for \(p=0.1\), \(k=1.36\) for \(p=0.05\), and \(k=1.63\) for \(p=0.01\). We therefore treat as independent the means of the four observations obtained from each subject.

  14. Although it makes no difference in practice, we further condition the distribution of \(r_{1}\) on the requirement that \(0\leq r_{1}\leq \frac 12\).

  15. Using different \(a\) and \(d\) for \(r_{1}\) and \(r_{2}\) has no significant effect.

  16. Recent working papers by Ahn et al. (2010) and Charness et al. (2012) report similar results with a different experimental design.

  17. The same goes for the minimax regret criterion, since this coincides with the maximin criterion in our experiment.

  18. For example, Keren and Gerritsen (1999) ask subjects to choose between betting on a red ball drawn from an urn which has a known probability of \(\frac {1}{3}\) of yielding a red ball and betting on a green ball or on a blue ball drawn from a different, ambiguous urn, in which green and blue together make up slightly more than \(\frac {2}{3}\) of the balls. They conclude that a large majority preference for betting on red is evidence of ambiguity aversion. Liu and Colman (2009) similarly use a one-choice setup. They offer subjects a choice between betting on a red ball drawn from an urn with a known probability of \(\frac 12\) of yielding a red ball and betting on a red ball drawn from an urn containing red and green balls in an unknown proportion. The prize from the uncertain urn was somewhat larger than the prize from the risky urn. They take a majority preference to bet on red from the risky urn to be evidence of ambiguity aversion.

  19. Genders were roughly equal in each case.

  20. These LOT-R results are consistent with the norms reported in Scheier et al. (1994).

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Authors and Affiliations

  1. Economics Department, University College London, London, WC1E 6BT, UK

    Ken Binmore

  2. Psychology Department, Australian National University, Canberra, ACT, 0200, Australia

    Lisa Stewart

  3. Department of Philosophy, Logic and Scientific Method, London School of Economics, London, WC2E 2AE, UK

    Alex Voorhoeve

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  1. Ken Binmore
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  2. Lisa Stewart
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  3. Alex Voorhoeve
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Correspondence to Alex Voorhoeve.

Additional information

Parts of the experimental data were gathered while Lisa Stewart was a researcher in the Harvard Psychology Department and Alex Voorhoeve was a Faculty Fellow at Harvard’s Safra Center for Ethics. We thank the Decision Science Laboratory at Harvard and the ELSE laboratory at University College London for the use of their facilities, and the Suntory and Toyota International Centres for Economics and Related Disciplines (STICERD) for financial support. Ken Binmore thanks the British Economic and Social Research Council through the Centre for Economic Learning and Social Evolution (ELSE), the British Arts and Humanities Research Council through grant AH/F017502 and the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC grant 295449. Alex Voorhoeve thanks the Safra Center for Ethics for its Faculty Fellowship and the British Arts and Humanities Research Council through grant AH/J006033/1. Results were presented at Bristol University, the European University Institute in Florence, Harvard University, the LSE, the University of Siena, and the University of York (UK). We thank Richard Bradley, Barbara Fasolo, Joshua Greene, Glenn Harrison, Jimmy Martinez, Katie Steele, Joe Swierzbinski, Peter Wakker, the Editors of and an anonymous referee for the JRU, and those present at our seminars for their comments.

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Binmore, K., Stewart, L. & Voorhoeve, A. How much ambiguity aversion?. J Risk Uncertain 45, 215–238 (2012). https://doi.org/10.1007/s11166-012-9155-3

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