Replies to Commentators

Abstract

I reply to two commentaries—one by Johanna Thoma and Jonathan Weisberg and one by James M. Joyce—concerning how risk-weighted expected utility theory handles the Allais preferences and Dutch books.

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Fig. 1

Notes

  1. 1.

    See Buchak (2013: 160–169).

  2. 2.

    See Briggs (2015) for a helpful example.

  3. 3.

    See Buchak (2015: 16–19).

  4. 4.

    Buchak (2013): 71.

  5. 5.

    See “Appendix 2”.

  6. 6.

    Another possibility that Thoma and Weisberg don’t explore is the possibility that variance increases as the amounts of money increase. This seems fairly plausible, since receiving $1 M or $5 M opens up a range of possibilities that weren’t present before. (And, again, the security added by the higher monetary value is security in the sense of eliminating or lowering the probability of the bad possibilities, not in the sense of narrowing the overall range of possibilities).

  7. 7.

    See “Appendix 3”. This was roughly the maximum variance that reconstructed the preferences given a 1.3:1 utility ratio and skew 5/−5.

  8. 8.

    See “Appendix 3”.

  9. 9.

    See MacCrimmon (1968), Moskowitz (1974), and Slovic and Tversky (1974).

  10. 10.

    See Moskowitz (1974) and Oliver (2003), respectively.

  11. 11.

    See also Moskowitz (1974), Slovic and Tversky (1974), and MacCrimmon and Larsson (1979).

  12. 12.

    20 of the participants violated EU-maximization, which Oliver notes is consistent with studies that have used monetary outcomes.

  13. 13.

    These reasons were also consistent with anticipated regret and anticipated disappointment; the main point is that they do not suggest probability weighting.

  14. 14.

    See especially Starmer (2000), who surveys the alternatives to EU-maximization and concludes that the evidence in favor of both probability weighting and loss aversion is strong, so that rank-dependent theories (including those that build in loss-aversion) are the most descriptively promising. For other surveys of alternatives to EU-maximization (including rank-dependent theories) and discussions of how these fit with empirical results, see Machina (1987), Camerer (1989), Sugden (2004), and Schmidt (2004). A different kind of result comes from an experiment by Abdellaoui et al. (2007), who asked individuals to determine their utility functions on the basis of introspection and also asked individuals to determine their preferences between particular gambles; the introspected utility functions agreed with those derived from rank-dependent utility theories but not with those derived from expected utility theory.

  15. 15.

    We also face the question of whether to count tendencies like those mentioned above as part of the normative or descriptive component. I argue that these should not be included in the normative component in Buchak (2013: 74–81).

  16. 16.

    See MacCrimmon (1968), Moskowitz (1974), and Slovic and Tversky (1974). However, MacCrimmon finds that after discussion with an experimenter, a substantial portion of subjects endorse the EU-conforming preferences and reasoning. See MacCrimmon (1968) for evidence that people do want to conform their preferences to transitivity.

  17. 17.

    Let i denote Jacob’s total fortune before considering any bets. REU({i + 1, 0.3; i + 0, 0.7}) = i + (0.3)2((i + 1) − i) = i + 0.09, and REU({i − 1, 0.3; i + 0, 0.7} = i − 1 + (0.7)2(i − (i − 1)) = i − 0.51. Thus, the value that each bet adds to the value of Jacob’s current holdings is 9¢ and −51¢, respectively.

  18. 18.

    The value that the bet adds to the value of Jacob’s current holdings is REU({i + 1 − 1, 0.3; i + 0 + 0, 0.7} − REU({i + 1, 0.3; i + 0, 0.7}) = i − (i + 0.09) = −0.09.

  19. 19.

    REU({i − 0.91, 0.3; i + 0.09, 0.7}) = (i − 0.91) + (0.7)2((i + 0.09) − (i − 0.91)) = i − 0.42.

  20. 20.

    REU({i + 0.91 − 0.91, 0.3; i − 0.09 + 0.09, 0.7} = REU({i, 0.3; i, 0.7}) = i.

  21. 21.

    See Buchak (2013: 205).

  22. 22.

    Buchak (2013: 211).

  23. 23.

    Here Joyce cites p. 211 of my book, the discussion of which concerns a case of trying to price the second bet while holding the first bet, rather trying to price the second bet after declining the first bet.

  24. 24.

    See Buchak (2013: 219-220) for a discussion of (and rejection of) myopia.

  25. 25.

    Buying only the first will be worth −1¢, buying only the second will be worth −1¢, buying neither will be worth 0¢, and buying both will be worth 40¢.

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Correspondence to Lara Buchak.

Appendix: Mathematica notebooks

Appendix: Mathematica notebooks

Appendix 1: Thoma and Weisberg’s original code

figurea

Appendix 2: Smaller utility ratio

figureb

Appendix 3: Skewness

figurec
figured

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Buchak, L. Replies to Commentators. Philos Stud 174, 2397–2414 (2017). https://doi.org/10.1007/s11098-017-0907-4

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Keywords

  • Decision theory
  • Risk
  • Risk-weighted expected utility
  • Rank dependence
  • Allais paradox
  • Dutch books