## 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.

This is a preview of subscription content, log in to check access.

## Notes

- 1.
See Buchak (2013: 160–169).

- 2.
See Briggs (2015) for a helpful example.

- 3.
See Buchak (2015: 16–19).

- 4.
Buchak (2013): 71.

- 5.
See “Appendix 2”.

- 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.
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.
See “Appendix 3”.

- 9.
- 10.
- 11.
- 12.
20 of the participants violated EU-maximization, which Oliver notes is consistent with studies that have used monetary outcomes.

- 13.
These reasons were also consistent with anticipated regret and anticipated disappointment; the main point is that they do not suggest probability weighting.

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

- 21.
See Buchak (2013: 205).

- 22.
Buchak (2013: 211).

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

- 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¢.

## References

Abdellaoui, M., Barrios, C., & Wakker, P. (2007). Reconciling introspective utility with revealed preferences: Experimental arguments based on prospect theory.

*Journal of Economics,**138,*356–378.Briggs, R. (2015). Costs of abandoning the Sure-Thing Principle.

*Canadian Journal of Philosophy,**45*(5–6), 827–840.Buchak, L. (2013).

*Risk and rationality*. Oxford: Oxford University Press.Buchak, L. (2015). Revisiting risk and rationality: A reply to Pettigrew and Briggs.

*Canadian Journal of Philosophy,**45*(5–6), 841–862.Camerer, C. F. (1989). An experimental test of several generalized utility theories.

*Journal of Risk and Uncertainty,**2,*61–104.Joyce, J. (2017). Commentary on Lara Buchak’s Risk and Rationality. doi:10.1007/s11098-017-0905-6.

MacCrimmon, K. R. (1968). Descriptive and normative implications of decision theory. In K. Borch & J. Mossin (Eds.),

*Risk and uncertainty*(pp. 3–23). New York: St. Martin’s Press.MacCrimmon, K. R., & Larsson, S. (1979). Utility theory: Axioms versus “paradoxes”. In M. Allais & O. Hagen (Eds.),

*Expected utility and the allais paradox*. Dordrecht: D. Reidel.Machina, M. (1987). Choice under uncertainty: Problems solved and unsolved.

*Journal of Economic Perspectives,**1*(1), 121–154.Moskowitz, H. (1974). Effects of problem representation and feedback on rational behavior in Allais and Morlat-Type Problems.

*Decision Sciences,**5,*225–242.Oliver, A. (2003). A quantitative and qualitative test of the Allais paradox using health outcomes.

*Journal of Economic Psychology,**24,*35–48.Savage, L. (1954).

*Foundations of statistics*. Dover: John Wiley and Sons.Schmidt, U. (2004a). Alternatives to expected utility: Formal theories. In S. Barberà, P. J. Hammond, & C. Seidl (Eds.),

*Chapter 15 of handbook of utility theory*(pp. 757–837). Boston: Kluwer Academic Publishers.Schmidt, U. (2004b). Alternatives to expected utility: Foundations. In S. Barberà, P. J. Hammond, & C. Seidl (Eds.),

*Chapter 14 of handbook of utility theory*(pp. 685–755). Boston: Kluwer Academic Publishers.Slovic, P., & Tversky, A. (1974). Who accepts Savage’s axiom?

*Behavioral Science,**19*(6), 368–373.Starmer, C. (2000). Developments in non-expected utility theory: The hunt for a descriptive theory of choice under risk.

*Journal of Economic Literature,**38,*332–382.Thoma, J., & Weisberg, J. (2017). Risk writ large. doi:10.1007/s11098-017-0906-5.

## Author information

### Affiliations

### Corresponding author

## Appendix: Mathematica notebooks

### Appendix: Mathematica notebooks

### Appendix 1: Thoma and Weisberg’s original code

### Appendix 2: Smaller utility ratio

### Appendix 3: Skewness

## Rights and permissions

## About this article

### Cite this article

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

Published:

Issue Date:

### Keywords

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