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When Allais meets Ulysses: Dynamic axioms and the common ratio effect

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Abstract

We report experimental findings about subjects’ behavior in dynamic decision problems involving multistage lotteries with different timings of resolution of uncertainty. Our within-subject design allows us to study violations of the independence and dynamic axioms: Dynamic Consistency, Consequentialism and Reduction of Compound Lotteries. We investigate the effects of changes in probability and outcome levels on the pattern of choices observed in the Common Ratio Effect (CRE) and in the Reverse Common Ratio Effect (RCRE) and on their dynamic counterparts. We find that the probability level plays an important role in violations of Reduction of Compound Lottery and Dynamic Consistency and the outcomes levels in violations of Consequentialism. Moreover, more than one quarter of our subjects satisfy the Independence axiom but violate two dynamic axioms. We thus suggest that there is a greater dissociation that might have been expected between preferences captured by dynamic axioms and those observed over single-stage lotteries.

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Fig. 1
Fig. 2

Notes

  1. The original parameters proposed by Allais are the following: x 1 = 5 M€, x 2 = 1 M€, q = 0.9 and r = 0.1.

  2. Logically, it means:

    • If RCL and CON hold then DC ⇔ IND. (Karni and Schmeidler 1991)

    • If RCL and DC hold then CON ⇔ IND. (Volij 1994)

    • If DC and CON hold then RCL ⇔ IND. (Volij 1994)

  3. The first experimental investigation of the decomposition of the independence axiom in a dynamic set up is due to Kahneman and Tversky (1979). Indeed, the isolation effect comes from the decomposition of IND between CON and DC+RCL.

  4. In this study, the authors refer to separability and timing independence for what we call consequentialism and dynamic consistency.

  5. Indeed, with P = (x 2; 1), Q = (x 1, 0; q), R = (0; 1) ∈ Lthese patterns imply that: ∃ P, Q, RL, ∃ r ∈ [0, 1] s.t. PQrP + (1 − r) RrQ + (1 − r)R which is the negation of IND which is formally stated as: ∀ P, Q, RL, ∀ r ∈ [0, 1], PQrP + (1 − r)RrQ + (1 − r)R. Subsequently, r is called ratio to recall the common ratio effect that contradicts this axiom (r = 0.1 in the original paradox of Allais (1953)).

  6. Consequentialism in the sense of Hammond is the conjunction of the two axioms we call CON and RCL.

  7. Laboratoire d’Economie Expérimentale de Montpellier (France)

  8. We ran 6 sessions of 19 participants each.

  9. For every problem type, participants were given instructions and a short questionnaire to check their understanding of the task.

  10. With the aim to study exclusively CRE violations of IND and of the dynamic axioms, it would have been more adequate to use ratio values between 0.1 and 0.5. This is exactly what is done in Nebout and Willinger (2013). However, in this study we are interested in both CRE and RCRE violations, so it is useful to have a ratio value over one half.

  11. Screenshots of each problem type are available in Appendix B (Figs. 3, 4, 5 and 6).

  12. We do not find such effect in our sample so we do not evoke this feature later in the paper.

  13. “Since the random lottery incentive system is widely used in experimental economics this points to a further motivation for testing dynamic choice principles. In any random lottery design, the subject makes precommitments to actions to be taken conditional on a chance event. Timing independence implies that these precommitments are in line with the actions which would be taken after the realisation of nature’s move. Separability implies that the latter actions are identical to those which would have been taken had the relevant decision problems been faced in isolation and for real. Thus, timing independence (DC) and separability (CON) are jointly sufficient for the validity of the random lottery incentive system.” Footnote 8 in Cubitt et al. (1998).

  14. This paucity of violations of dominance is re-assuring and suggests that a majority of subjects took the tasks seriously, despite hypothetical incentives.

  15. S 1: χ 2 = 0.157 p-value = 0.692, S 2: χ 2 = 1.130 p-value = 0.288, S 3: χ 2 = 0.792 p-value = 0.374 and S 4: χ 2 = 0.577 p-value = 0.448

  16. This case, where the risky option is chosen in the scale up problem and the safe option in the scale down problem, has been accounted for theoretically by Blavatskyy (2010).

  17. Based on Table 4 for proportions and Table 13 in Appendix A for chi-square values:

    • CON is rejected (both criteria pooled) in 37.75% of the cases for r L and 40.00% for r H. This difference is not significant (χ 2 = 0.143 p-value = 0.705). When we aggregate over the ratios in Table 4, CON is rejected also in about 40% of the cases (more precisely 39.44% and 39.26% respectively for x L and x H, this difference being not significant, χ 2 = 0.002 p-value = 0.963).

    • DC is rejected (both criteria pooled) in 35.06% of the cases for x L and 37.78% for x H, which is not significantly different (χ 2 = 0.306 p-value = 0.580). DC is more frequently rejected for r L (40.40%) than for r H (34.86%), but the difference is not statistically significant (χ 2 = 1.188 p-value = 0.276).

    • RCL is rejected in 39.86% of the cases for r L and 41.22% for r H, a non-significant difference (χ 2 = 0.117 p-value = 0.733). The difference of rejection frequency between x L and x H (i.e. when we aggregate over the ratios) is also non-significant (39.19% for x L and 41.89% for x H, χ 2 = 0.565 p-value = 0.452).

  18. It is also true for each outcome level, for x 1 = 20 χ 2 = 0.052 p-value = 0.820, for x 1 = 24 χ 2 = 0.003 p-value = 0.959, for x 1= 80 χ 2= 0.171 p-value = 0.679 and for x 1 = 95 χ 2 = 0.105 p-value = 0.746

  19. As a reminder, these sure gains are x 2 = 15€ for x L and x 2 = 60€ for x H.

  20. Mc Nemar change test, x 1 = 20: χ 2 = 0.750 p-value = 0.387, x 1 = 24: χ 2 = 0 p-value = 1, x 1 = 80: χ 2 = 0.842 p-value = 0.359, x 1 = 95: χ 2 = 1.389 p-value = 0.239

  21. Mc Nemar change test, x 1 = 20: χ 2 = 3.125 p-value = 0.077, x 1 = 24: χ 2 = 1.777 p-value = 0.182, x 1 = 80: χ 2 = 1.388 p-value = 0.239 and x 1 = 95: χ 2 = 0.100 p-value = 0.752

  22. Note that this result is presented in terms of RCRE because, for RCL, this type of violation is more frequently observed than CRE.

  23. This difference is significant at an aggregate level (χ 2 = 30.392, p-value < 0.001). At the aggregated outcome level this difference is also significant (for x L, χ 2 = 8.013 p-value = 0.005 and for x H, χ 2 = 22.919 p-value < 0.001). Finally this result is also obtained for all outcome levels except for x 1 = 20, (for x 1 = 20: χ 2 = 1.546 p-value = 0.214, for x 1 = 24: χ 2 = 6.266 p-value = 0.012, for x 1 = 80: χ 2 = 19.151 p-value < 0.001 and for x 1 = 95: χ 2 = 4.5432 p-value = 0.033).

  24. We thank an anonymous referee for suggesting that we explore this dimension of our data set and that we run this fruitful analysis.

  25. More precisely, 72.22% of U 2 / D 3 and 27.78% of D 3 / U 2 are observed for low outcomes, while the frequencies are respectively 14.63% and 85.37% for high outcomes and the difference is statistically significant (χ 2(1) = 23.857 p-value < 0.001). The difference is also significant for each ratio level (r L : χ 2(1) = 4.861 p-value = 0.027, r H : χ 2(1) = 17.202 p-value < 0.001). If we aggregate the outcome levels and test for the effects of the ratio, the frequencies of observation of (U 2 / D 3, D 3 / U 2) for r L and r H are respectively (25%, 75%) and (51.02%, 48.98%), a significant difference (χ 2(1) = 3.954 p-value = 0.047).

  26. For a discussion of the differences between within- and between-subjects experiments, see for example Ballinger and Wilcox (1997).

  27. Nebout and Willinger (2013) use real incentives (RIS) and collected data that, for DC and CON, are consistent with the one presented in this paper.

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Acknowledgments

This research was funded by CNRS, ANR Risk Attitude and University of Montpellier. We are grateful to Mohammed Abdellaoui, Thomas Epper, Brian Hill, John Hey, Chris Starmer, Peter Wakker and Marc Willinger, to participants in conferences in Lyon, Dijon, Barcelona and Montpellier and in seminars in Queensland University of Technology and Monash University for helpful comments. We also thank the editor and an anonymous referee for their constructive comments on earlier drafts of this paper and the managing editor, Christina Stoddard, for her great help through the entire publication process.

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Correspondence to A. Nebout.

Appendices

Appendix A: Additional statistics

Table 12 Chi-square values of association between Accept/Reject variables
Table 13 Chi-square values for Accept/Reject variables depending of the parameter levels
Table 14 Chi-square values of the independence tests between CRE and RCRE for different parameter levels
Table 15 Frequency and percentage (in brackets) of each profile for 2-switches subjects
Table 16 Chi-square values for association between axiom’s rejection variables without 3-switches type

Appendix B: Screenshots

Fig. 3
figure 3

Scaled up problem (S1)

Fig. 4
figure 4

Scaled down problem (S4)

Fig. 5
figure 5

Two stages problem (S3)

Fig. 6
figure 6

Prior lottery problem (S2)

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Nebout, A., Dubois, D. When Allais meets Ulysses: Dynamic axioms and the common ratio effect. J Risk Uncertain 48, 19–49 (2014). https://doi.org/10.1007/s11166-014-9184-1

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