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Subjective Bayesian beliefs

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Abstract

A large literature suggests that many individuals do not apply Bayes’ Rule when making decisions that depend on them correctly pooling prior information and sample data. We replicate and extend a classic experimental study of Bayesian updating from psychology, employing the methods of experimental economics, with careful controls for the confounding effects of risk aversion. Our results show that risk aversion significantly alters inferences on deviations from Bayes’ Rule.

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Notes

  1. The most unlikely pattern is (5,0) with probability of occurring equal to 0.044, followed by the pattern (7,2) with probability 0.091, and the pattern (11,6) with probability 0.104. Based on these probabilities we chose the frequency of each sample size to roughly equalize the expected frequency of the most unlikely patterns in each session.

  2. For example, each subject filled out 30 betting sheets, such as the one shown by Table 1. At the end of the experiment we first randomly chose, for each subject, one of these betting sheets, and then we chose 1 of the 19 bookies within that selected sheet. If the subject placed the allocated £3 on the box that was actually chosen, he was paid the amount that corresponds to the odds offered by that bookie.

  3. We provide a complete derivation of the application of Bayes’ Rule for this process in Appendix 2, which can be found in the accompanying online document.

  4. In our lottery experiments the subjects are told at the outset that any expression of indifference would mean that the experimenter would toss a fair coin to make the decision for them if that choice was selected to be played out. Hence one can modify the likelihood to take these responses into account either by recognizing this is a third option, the compound lottery of the two lotteries, or alternatively that such choices imply a 50:50 mixture of the likelihood of choosing either lottery, as illustrated by Harrison and Rutström (2008; p.71). We do not consider indifference here because it was an extremely rare event.

  5. The normalizing term ν is defined as the maximum utility over all prizes in this lottery pair minus the minimum utility over all prizes in this lottery pair, and ensures that the normalized EU difference [(EUR - EUL)/ν] remains in the unit interval. As μ → ∞ this specification collapses ∇EU to 0 for any values of EUR and EUL, so the probability of either choice converges to ½. So a larger μ means that the difference in the EU of the two lotteries, conditional on the estimate of r, has less predictive effect on choices. Thus μ can be viewed as a parameter that flattens out, or “sharpens,” the link functions implicit in (4). This is just one of several different types of error story that could be used, and Wilcox (2008) provides a masterful review of the implications of the strengths and weaknesses of the major alternatives.

  6. The normalizing term ν is given by the value of r and the lottery parameters, which are part of X.

  7. This qualitative result is exactly the same as one finds using a Quadratic or Linear scoring rule (providing, in the latter case, that one does not go to the extreme of exact risk neutrality): see Andersen et al. (2014).

  8. Holt and Smith (2009; §5) make exactly the same observation in a comparable study that we discuss further in Section 5.

  9. In its traditional form the RDU model assumes reduction of compound lotteries, a point we return to below. Our application here is unconventional in the application of concepts of “probabilistic sophistication” to a particular non-EUT model, in the spirit of Machina and Schmeidler (1992, 1995) and Grant (1995). It is unconventional in the sense of assuming that individuals behave as if they distort (weight) objective probabilities, but they are otherwise probabilistically sophisticated.

  10. In many physical processes, for example, threshold effects can lead to extreme outcomes rather than intermediate ones. So one person might believe the threshold has been exceeded, and another person might believe it has not.

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Acknowledgments

Harrison thanks the U.S. National Science Foundation for research support under grants NSF/HSD 0527675 and NSF/SES 0616746, and we thank the referee and seminar and conference participants for valuable comments.

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Authors and Affiliations

  1. Warwick Business School, University of Warwick, Coventry, UK

    Constantinos Antoniou & Daniel Read

  2. Department of Risk Management & Insurance and Center for the Economic Analysis of Risk, Robinson College of Business, Georgia State University, Atlanta, GA, USA

    Glenn W. Harrison

  3. Durham Business School, Durham University, Durham, UK

    Morten I. Lau

  4. Copenhagen Business School, Copenhagen, Denmark

    Morten I. Lau

  5. School of Economics, University of Capetown, Capetown, South Africa

    Glenn W. Harrison

  6. IZA- Institute for the Study of Labor, Bonn, Germany

    Glenn W. Harrison

Authors
  1. Constantinos Antoniou
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  2. Glenn W. Harrison
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  3. Morten I. Lau
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  4. Daniel Read
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Corresponding author

Correspondence to Glenn W. Harrison.

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Antoniou, C., Harrison, G.W., Lau, M.I. et al. Subjective Bayesian beliefs. J Risk Uncertain 50, 35–54 (2015). https://doi.org/10.1007/s11166-015-9208-5

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