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Is the Allais paradox due to appeal of certainty or aversion to zero?

Abstract

We provide a novel but intuitive explanation for expected utility violations found in the Allais paradox: individuals are commonly averse to receiving nothing. We call this phenomenon the zero effect. Our laboratory experiments show support for the zero effect. By contrast, the evidence for the certainty effect is weak to nonexistent.

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Notes

  1. For example, a similar experiment in Kahneman and Tversky (1979) finds that 12 out of 72 participants chose A over B, but 59 out of 72 chose \(A'\) over \(B'\). These numbers imply that at most 35% chose either both A and \(A'\) or both B and \(B'\), with the remainder violating expected utility theory.

  2. In our extension to the common ratio effect (CRE) below in Sect. 4, we give the formal definition of the CRE due to Battalio et al. (1990).

  3. For approaches staying within a triangle design that provide indirect tests by considering multiple triangles, see Appendix B.

  4. Although allowing for indifference adds a complication, we permit it to rule out indifference as a cause of Allais-type behavior. See Harrison (1994).

  5. We deliberately kept the differences in the expected values of the lotteries small, as it is known that participants facing large differences in expected real payments in common consequence tasks overwhelmingly choose the risky lottery (see Conlisk 1989; Fan 2002).

  6. If we include indifferences, there are five additional patterns of interest: IIIIII (expected utility); RRIRRR and IISIII (certainty effect), and ISSSSS and RIIIII (zero effect). We run our tests both including observations with indifferences and excluding observations with indifferences.

  7. We manipulate the presentation and the incentive structure to improve the generalizability of our results rather than to test their direct effects. See also Moskowitz (1974), Keller (1985), Gottlieb et al. (2007), Conlisk (1989), Beattie and Loomes (1997), Fan (2002), Blavatskyy et al. (2020).

  8. Littenberg et al. (2003) establish the reliability of pencil-and-paper instruments for tasks involving decisions under risk.

  9. Our null hypotheses are that each predicted pattern occurs no more than due to chance, i.e., with probability \(1/64\). We also tested the non-EU patterns to see if they occur no more than due to chance conditional on non-EU choice, i.e., with probability \(1/62\). The results were the same.

  10. To elaborate, we note that there are additional patterns consistent with the CCE, aside from the certainty effect and zero effect patterns. Recall from Definition 1 that the CCE depends only on decisions under two common consequences, \(c\in \{0,m\}\). Therefore, any pattern with riskier choice when c=0 than when c=m would be consistent with the CCE (e.g. \(R\_S\_\_\_\)).

  11. We also consider numerous alternative tests, allowing for heterogeneity in trembles across common consequences, in the spirit of the true-and-error model of Birnbaum and Schmidt (2008), and tests that can be considered a simple version of the true-and-error model as in Loomes et al. (1991). In addition, we modified our main tests to include other forms of heterogeneity in trembles, such as errors that depend on presentation order or presentation style. We also allowed for individual heterogeneity in errors based on preferences, e.g., allowing for an EU type who prefers a safe lottery to have different rates of trembles than an EU type who prefers a risky lottery. None of these variations changed our main conclusions (details available from the authors on request).

  12. Tables 4 and 5 show the results with indifferences excluded. We also analyzed the data including indifferences. The results do not change.

  13. We use the bootstrap method to estimate the standard error of the estimated parameters. For the bootstrap estimation, we re-sampled the experimental data 10,000 times.

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Acknowledgements

Thanks to John Duffy and Linda Moya, who made it possible for us to run our experiment. Many people provided us with helpful comments. Thanks especially to Dave Dillenberger, Coty Gonzales, Ed Green, Yusufcan Masatlioglu, Charles Noussair, Timonthy Shields, Peter Wakker, and workshop participants at the FUR conference, the Center for Behavioral Decision Research workshop at Carnegie Mellon, and the Society for the Advancement of Economic Theory conference. This paper originated in discussions with Horacio Arlo-Costa, and we dedicate it to his memory.

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Correspondence to Elif Incekara-Hafalir.

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Incekara-Hafalir, E., Kim, E. & Stecher, J.D. Is the Allais paradox due to appeal of certainty or aversion to zero?. Exp Econ 24, 751–771 (2021). https://doi.org/10.1007/s10683-020-09678-4

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Keywords

  • Allais paradox
  • Certainty effect
  • Common consequence
  • Common ratio
  • Decision theory
  • Zero effect