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Reconsidering the common ratio effect: the roles of compound independence, reduction, and coalescing

Abstract

Common ratio effects should be ruled out if subjects’ preferences satisfy compound independence, reduction of compound lotteries, and coalescing. In other words, at least one of these axioms should be violated in order to generate a common ratio effect. Relying on a simple experiment, we investigate which failure of these axioms is concomitant with the empirical observation of common ratio effects. We observe that compound independence and reduction of compound lotteries hold, whereas coalescing is systematically violated. This result provides support for theories which explain the common ratio effect by violations of coalescing (i.e., configural weight theory) instead of violations of compound independence (i.e., rank-dependent utility or cumulative prospect theory).

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Notes

  1. Allais (1953a, b, 1979a, b) and Morlat (1953) were the first to develop lotteries which revealed that subjects violate the axioms of expected utility, primarily the independence axiom and the sure-thing principle. Allais (1979b, p. 533) reported: “During the 1952 Paris Colloquium, I had Savage respond over lunch to a list of some 20 questions. His answers to each were incompatible with the basic axioms of his own theory.” In the second edition of his Foundations of Statistics, Savage (1972, p. 103) explained that his immediate reaction was based on error. He argued that a different presentation of Allais’ lotteries as shown in his Table 1 would not have trapped him in his immediate error.

    For other literature on such anomalies which were summarized under the headlines common ratio effect and common consequence effect, see MacCrimmon (1967), Morrison (1967), Moskowitz (1974), Slovic and Tversky (1974), MacCrimmon and Larsson (1979), Kahneman and Tversky (1979), Conlisk (1989), MacDonald and Wall (1989), Battalio et al. (1990, p. 37), Starmer and Sugden (1991, 1993), Harless (1992a, b), Harrison (1994, p. 231) Burke et al. (1996), Beattie and Loomes (1997), Cubitt et al. (1998a), Fan (2002), Birnbaum (2004), and Blavatskyy (2010).

  2. The common ratio effect was explained by way of the fanning-out hypothesis in a Marschak-Machina triangle. In particular, it is observed in the bottom right corner of the Marschak-Machina triangle. Some observations located it at the boundary of a Marschak-Machina triangle. This comes up to the certainty effect, which results from overweighing payoffs which are obtained with certainty; see Kahneman and Tversky (1979, p. 265), Tversky and Kahneman (1986, p. S266), and Conlisk (1989, p. 397). Having investigated both explanations, Conlisk (1989, p. 401) concluded: “The direction of the systematic effect favors the certainty effect hypothesis over at least the linear version of the fanning-out hypothesis.” Our paper shows that the common ratio effect is associated with coalescence.

  3. In his Gains-Decomposition-Utility (GDU) model, Luce (2000, pp. 200–202) proposed to arrange multiple-branch decision problems in terms of multiple-stage trees.

  4. Kahneman and Tversky (1979, p. 271) argue: “In order to simplify the choice between alternatives, people often disregard components that the alternatives share, and focus on the components that distinguish them ... This approach to choice problems may produce inconsistent preferences, because a pair of prospects can be decomposed in more than one way, and different decompositions sometimes lead to different preferences. We refer to this phenomenon as the isolation effect.”

    If one of the second-stage lotteries has a payoff under certainty, we encounter a particular species of the isolation effect, viz. the pseudocertainty effect. This terminology was coined by Tversky and Kahneman (1986, pp. S267–S268), “because an outcome that is actually uncertain is weighted as if it were certain.”

  5. It seems that Smith’ (1982) precepts for orderly experiments had triggered a heated debate about financial incentives, and, among them, about the random-lottery incentive system. Comprehensive investigations by Camerer and Hogarth (1999) for 74 experiments showed that the superiority of financial incentives is not apodictic. Rather it depends on the subject matter of the respective experiments and the particular circumstances on whether financial or hypothetical payoffs are superior. Camerer (1995, p. 635) remarked:

    The effect of paying subjects is likely to depend on the task they perform. In many domains, paid subjects probably do exert extra mental effort, which improves their performance, but in my view choice over money gambles is not likely to be a domain in which effort will improve adherence to rational axioms. Subjects with well-formed preferences are likely to express them truthfully, whether they are paid or not.

    Davis and Holt (1993, p. 450) tend to endorse financial incentives. Even Smith and Walker (1993, p. 246), who carefully surveyed 31 experimental studies, are less apodictic than Smith (1982) in unconditionally endorsing financial incentives: “neither of the polar views—only reward matters or reward does not matter—are sustainable across the range of experiments.”

  6. Beattie and Loomes (1997, p. 158) rightly remark that Harrison’s “sample sizes are too small for statistical tests to discriminate between the differences.”

  7. At the time of the experiment one euro (€) amounted to some $1.30. Our payments might appear moderate but €4.00 in ten minutes comes up to €24.00 per hour. When students job in Germany, they can usually make €10.00 per hour. Moreover, we distributed our questionnaires during undergraduate classes so that students had neither travel expenditures nor costs of time involved with the experiment. Since we addressed only economics classes, the majority of students showed great interest in economic experiments.

  8. We used this arrangement instead of differently colored marbles, as Birnbaum (2004) did, to avoid framing effects, although they are minor as evidenced by Birnbaum (2004, p. 99).

  9. As for respective methods, see, e.g., the impressive experiment carried out by Holt and Laury (2002). They started asking subjects to choose a lottery from the pair \(L_1=(\$2.00,0.1;\$1.60,0.9), \; G_1=(\$3.85,0.1;\$0.10, 0.9)\). Obviously \(L_1\) will be chosen. Then they increased the probability of gaining the higher payoff in both lotteries stepwise by 10 percent and decreased the probability of gaining the lower payoff by 10 percent, ending with \(L_{10}=(\$2.00,1;\$1.60,0), \; G_{10}=(\$3.85,1;\$0.10, 0)\). Obviously \(G_{10}\) will be chosen. A risk-neutral subject will choose \(L\) four times before switching to \(G\). Greater lottery indices of the switching point indicate greater risk aversion. For a more refined approach along these lines cf. Harrison et al. (2007, pp. 88–92). A similar method was earlier employed by Loomes and Sugden (1998). They arranged model lotteries taking expected utility as their core model in a series of Marschak-Machina triangles so that their gradients steadily increased rendering the riskier lotteries more attractive. Then they used the reversal frequencies as indicators of risk preferences.

    Curiously enough, although these experiments could have been used to check homogeneity of risk preferences of subjects in different groups, to the best of our knowledge this research was never used for this purpose. Rather it was carried out under the implicit assumption that risk preferences were the same across the group of all subjects used for these experiments. The aim of this research was to investigate how risk preferences vary as the payoffs used in the respective experiments are scaled up (cf. also Footnote 14).

  10. Such distortions have been recently reported by Cox et al. (2014a), Cox et al. (2014b), and Harrison and Swarthout (2012). For contrary results see Starmer and Sugden (1991), Beattie and Loomes (1997), Cubitt et al. (1998b), and Hey and Lee (2005).

  11. Note that the following four experiments are based on the experimental design of Kahneman and Tversky (1979, p. 266, Problems 3 and 4, and p. 271, Problem 10). We divided their payoffs, which were stated in contemporary Israeli Pounds (at that time, the median net monthly income of a family was 3,000 Israeli Pounds) by 1,000 to receive the payoffs of our experiments in terms of €. The original paper by Allais (1953b, p. 527) covered only the common consequence effect, not the common ratio effect.

    Our experiment differs from the Kahneman and Tversky (1979) experiment in three important aspects: first, instead of presenting the probabilities straight in terms of numbers, we presented them in terms of draws of a marble from an urn. Carlin (1990) showed that this seemingly minor move has marked effects. Second, we used real rather than hypothetical payoffs. Third, we analyze the role of event-splitting effects.

  12. We also conducted a \(\chi ^2\) test (not reported here) which (as a two-sided test) fully confirmed the results of Table 3.

  13. Note that, as translated into our experiment, Cubitt et al. (1998a, p. 1375) observed that 38 % of their subjects chose Option B in Experiment 1 and 48.1 % chose Option B in Experiment 3. Obviously the majority of their subjects appreciated a 80 % chance of getting £16 more than a certain payoff of £10, whereas only 51.22 % of our subjects considered a 80 % chance of getting € 4 as preferable to a certain payoff of € 3. This difference is well explained by the different level and spread of rewards. Note, however, that the response pattern of subjects is similar: whereas Cubitt et al. (1998a) observed 27 % more B responses in Experiment 3 than in Experiment 1, our figures amount to 43 % more B responses (see also Cubitt et al. (1998a, p. 1376, H11–H12)).

  14. Holt and Laury (2002, pp. 1648–50) observed major increases in risk aversion as the real payoffs of their model lotteries are scaled up. A follow-up experiment by Harrison et al. (2005) controlling for order effects confirmed Holt and Laury’s results, but at a lower level of risk aversion. For further follow-up work see Holt and Laury (2005). Smith and Walker (1993, p. 259) found that real rewards reduce the variance of data around the predicted outcome.

  15. Note that there is a major difference between our results and the Cubitt et al. (1998a) results. Whereas we observe nearly 50 % choices of Option B in Experiments 1 and 4, Cubitt et al. (1998a, p. 1375) (as translated into our experiments) observe 38 % choices of Option B in Experiment 1 and 66% in Experiment 4. We have no explanation for the marked prevalence of risky choices in Experiment 4 in the Cubitt et al. (1998a) experiment.

  16. Seidl (2013) showed that probability weighing may, equivalently to payoff weighing, be used to “solve” or regain St. Petersburg paradoxes.

  17. Cf., e.g., Birnbaum et al. (1971, 1992), Birnbaum (1974), Birnbaum and Stegner (1979), and Birnbaum and Chavez (1997).

  18. Note that decision models based on weighed cumulative probabilities (e.g., rank dependent and rank and sign dependent utility theories as developed by Quiggin 1982, 1985, 1993; Luce and Fishburn 1991, 1995, and Tversky and Kahneman 1992) cannot deal with event splitting because identical terms of cumulative probabilities for the same events cancel. See also Birnbaum (2008).

  19. Seminal work on probability weighing functions was done by Prelec (1998) and Gonzalez and Wu (1999).

  20. As a résumé of their work, Starmer and Sugden (1993, p. 253) remarked: “Perhaps the most significant feature of our results is that they provide evidence of event-splitting effects that are inconsistent with almost all current theories of choice under [un]certainty.” Our research has hopefully added another piece of evidence on event-splitting effects.

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Acknowledgments

We are indebted to Gerd Hansen, Jan Krause, and to two anonymous referees for helpful comments. Remaining errors are ours.

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Schmidt, U., Seidl, C. Reconsidering the common ratio effect: the roles of compound independence, reduction, and coalescing. Theory Decis 77, 323–339 (2014). https://doi.org/10.1007/s11238-014-9456-x

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Keywords

  • Common ratio effect
  • Coalescing
  • Reduction
  • Compound independence
  • Event splitting
  • Branch splitting
  • Isolation effect
  • Allais paradox

JEL Classifications

  • C91
  • C44
  • D81