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Rationality on the rise: Why relative risk aversion increases with stake size

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Abstract

How does risk tolerance vary with stake size? This important question cannot be adequately answered if framing effects, nonlinear probability weighting, and heterogeneity of preference types are neglected. We show that the observed increase in relative risk aversion over gains cannot be captured by the curvature of the value function. Rather, it is predominantly driven by a change in probability weighting of a majority group of individuals who weight probabilities of high gains more conservatively. Contrary to gains, no coherent change in relative risk aversion is observed for losses. These results not only challenge expected utility theory, but also prospect theory.

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Notes

  1. Arguably, game shows provide a decision environment which differs radically from everyday situations. Contestants face a once-or-never opportunity to win an extremely high amount of money, and they have to take their decisions under time pressure and under the scrutiny of a large audience. Estimates of overall risk aversion in game shows suggest a relatively high level of risk tolerance which is most likely not representative of risk attitudes in “normal” situations. Nevertheless, in line with the experimental evidence, when facing comparatively higher stakes contestants become relatively more risk averse.

  2. Instructions are available upon request.

  3. At the time of the experiment one Chinese Yuan equaled about 0.12 U.S. Dollars (USD). Low-stake outcomes ranged from 0.48 to 6.6 USD, high-stake outcomes from USD 7.8 to 114 USD.

  4. Advantages and potential drawbacks are discussed in Andersen et al. (2006a).

  5. To see this, assume v(x) = − λ( − x)β for losses. For nonmixed loss lotteries the parameter of loss aversion λ cancels out in the definition of the certainty equivalent ce: \(-\lambda(-ce)^\beta=-\lambda(- x_1)^\beta w(p) -\lambda (-x_2)^\beta(1-w(p))\) holds for any arbitrary value of λ.

  6. A necessary and sufficient preference condition for this specification is presented in Gonzalez and Wu (1999).

  7. Moreover, the function generally fits equally well as the two-parameter function developed by Prelec (1998).

  8. As the graphs in Appendix B show, the distributions of the estimated ξ i clearly depart from normality.

  9. As all the lotteries under consideration here are single-domain ones with a maximum of two non-zero outcomes only, there seems to be no need for modeling any lottery-specific errors, such as heteroskedasticity due to the position of the lottery in the sequence of choices, in addition to range dependence and domain dependence. The approach adopted in this paper is to characterize average behavior of large groups of people. Any sequence effects at the individual level disappear at the aggregate level if the order of the lotteries is individually randomized.

  10. The estimates of the model viewing types as domain specific are presented in Appendix A.

  11. The curvature for low-stake gains is estimated to be 0.467, in line with numerous previous findings (Stott 2006). The estimate for β 0 amounts to 1.165, indicating slight concavity of the value function. However, its curvature is not statistically distinguishable for linearity. That the value function for losses is not convex, as predicted by prospect theory, is not an unusual finding (Abdellaoui 2000; Bruhin et al. 2007).

  12. The study on risky gains by Kachelmeier and Shehata (1992), conducted in Beijing as well, finds that stake size interacts significantly with probability level, which is in line with our findings, but their data set is not sufficiently rich to draw any conclusions on relative contributions of outcome valuation and probability weighting. Furthermore, observed certainty equivalents in our data set show a clearly defined fourfold pattern of risk attitudes for both low stakes and high stakes, whereas Kachelmeier and Shehata find practically no risk aversion in choices over low stakes. The authors attribute this lack of risk aversion to the specifics of their elicitation procedure: Certainty equivalents were elicited as minimum selling prices, which seems to have induced some kind of loss aversion in subjects’ responses.

  13. “The problem of identifying the number of classes is one of the issues in mixture modeling with the least satisfactory treatment.” (Wedel 2002, p. 364).

  14. Results are available upon request.

  15. Similarly to our findings, the delay dependence of risk tolerance is solely due to a change in probability weights whereas delay has no discernible effect on the valuation of monetary outcomes.

  16. Such an interaction effect is not modeled by affective utility theories, as intensity of anticipated emotions is assumed to be independent of lottery characteristics.

  17. In principle, the framework of affective utility could also account for the absence of a stake effect for losses as well. Our Chinese subjects exhibit comparatively strong elation proneness over low-stake gains which diminishes considerably for high-stake gains. In the domain of low-stake losses elation seeking and disappointment aversion are more equally balanced in the first place and, therefore, there may be less leeway for decreasing elation proneness further. Note that the probability weighting curves for losses closely resemble the curve for high-stake gains. Of course, this conjecture is purely speculative.

  18. Abdellaoui and Bleichrodt (2007) tested Gul’s theory of disappointment aversion and found it to be too parsimonious to explain their data.

  19. The procedure is written in the R environment (R Development Core Team 2006).

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Acknowledgements

We are particularly thankful to Mei Wang, Xiaofei Xie, Marc Schuerer, Ernst Fehr, George Loewenstein, and Roberto Weber, to the participants of the ESA World Conference 2007, the EEA-ESEM 2008 and the Behavioral Decision Research Seminar at Carnegie Mellon University, as well as to an anonymous referee. The usual disclaimer applies. This research was supported by the Swiss National Science Foundation (Grant 100012-109907).

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Correspondence to Helga Fehr-Duda.

Appendices

Appendix A: Domain-specific behavior

The main goal of estimating a finite mixture model is to classify each individual to one of C different behavioral types. Our econometric model, as specified in Eq. 8, requires that each individual belongs to only one behavioral type, even though the behavioral parameters of each type are domain specific and may be different for decisions framed as gains or as losses. This makes the model very flexible in classifying the individuals into types which may exhibit domain-specific behavioral patterns, but retains the individual as unit of classification.

However, one can also estimate the finite mixture model separately for each domain. In such a case, behavioral types are only characterized within their specific domain and, consequently, each individual will be classified into two types, one for decisions framed as gains and the other for decisions framed as losses. The (overall) log likelihood of such a specification,

$$ \begin{array}{lll} \ln \tilde{L}\big(\tilde{\Psi}; ce, \mathcal{G}\big) &=& \label{equ:domain} \left[ \sum_{i=1}^{N}{ \ln \sum_{c=1}^{C} \tilde{\pi}_c^{(w)} f\left({ce_i^{(w)}, \mathcal{G}^{(w)}; \tilde{\theta}_c^{(w)},\tilde{\xi}_i^{(w)}}\right)} \right]\\ &&+ \left[ \sum_{i=1}^{N}{ \ln \sum_{c=1}^{C} \tilde{\pi}_c^{(l)} f\left({ce_i^{(l)}, \mathcal{G}^{(l)}; \tilde{\theta}_c^{(l)},\tilde{\xi}_i^{(l)}}\right)} \right] \, , \end{array} $$
(9)

is completely separable into decisions framed as gains, (w), and decisions framed as losses, (l). Note that since the above model (9) does not nest the original model (8) we cannot test these two specifications against each other. Nevertheless, as Table 4 shows, the separated model achieves a lower log likelihood than the original model, the estimates of which are displayed in Table 2, even though it has one more parameter. It also results in a higher value of the BIC (−62,058 vs. −62,163). This shows that the above specification is clearly inferior to domain-independent typing.

Table 4 Classification of behavior: domain-dependent types

At the level of individual type assignment, the main difference between the domain-specific model and the domain-independent one is that 13 people, identified as overall EUT type, move to the Non-EUT group for gains, which explains the lower percentage of EUT types in the domain-specific model (19.5% vs. 26.6%).

Appendix B: Error parameter distribution

Fig. 8
figure 8

Distributions of estimated \(\hat{\xi_i}\) by domain

Appendix C: Estimation of the finite mixture regression model

As it is generally the case in finite mixture models, direct maximization of the log likelihood function

$$ \ln {L} \big(\Psi; ce, \mathcal{G} \big) = \sum_{i=1}^{N}{ \ln \sum_{c=1}^{C}{\pi_c \, f\big(ce_i, \mathcal{G}; \theta_c, \xi_i\big)} } $$

may encounter several problems, even if it is in principle feasible (for a general treatise see for example McLachlan and Peel (2000)). First, the highly non-linear form of the log likelihood causes the optimization algorithm to be rather slow or even incapable of finding the maximum. Second, the likelihood of a finite mixture model is often multimodal and, therefore, we have no guaranty that a standard optimization routine will converge towards the global maximum rather than to one of the local maxima.

However, if individual group-membership were observable and indicated by t ic  ∈ {0,1} the individual contribution to the likelihood function would be given by

$$ \tilde{\ell}\big(\Psi_i; ce_i, \mathcal{G}, t_i \big) = \prod_{c=1}^{C} {\left[ \pi_c \, f\big(ce_i, \mathcal{G}; \theta_c, \xi_i \big) \right]^{t_{ic}}} $$

By using the above formulation and taking logarithms, the complete-data log likelihood function

$$ \ln { \tilde{L} } \big(\Psi; ce, \mathcal{G},t \big) = \sum_{i=1}^{N}{ \sum_{c=1}^{C}{t_{ic} \left[\ln{\pi_c} + \ln{f\big(ce_i, \mathcal{G}; \theta_c, \xi_i\big)} \right]} } $$

would follow directly. As relative group sizes sum up to one, their maximum likelihood estimates, \(\hat{\pi}_c=1/N \sum_{i=1}^{N}{t_{ic}}\), would be given analytically by the relative number of individuals in the respective group. Furthermore, the maximum likelihood estimates of the group-specific parameters could be obtained separately in each group by numerically maximizing the corresponding joint density function which would simplify the optimization problem considerably.

The EM algorithm proceeds iteratively in two steps, E and M, while it treats the unobservable t ic as missing data. In the E-step of the (k + 1)-th iteration the expectation of the complete-data log likelihood \(\tilde{L}\), given the actual fit of the data Ψ (k), is computed. This yields, according to Bayes’ law, the posterior probabilities of individual group-membership

$$ \tau_{ic}\left(ce_i, \mathcal{G}; \Psi_i^{(k)} \right)= \frac{\pi_c^{(k)} \, f\left(ce_i, \mathcal{G}; \theta_c^{(k)}, \xi_i^{(k)} \right)} {\sum_{m=1}^{C}{\pi_m^{(k)} \, f\left(ce_i, \mathcal{G}; \theta_m^{(k)}, \xi_i^{(k)} \right)}} $$

which replace the unknown indicators of individual group-membership, t ic .

Given \(\tau_{ic}\left(ce_i, \mathcal{G}; \Psi_i^{(k)} \right)\), the complete-data log likelihood, \(\tilde{L}\), is maximized in the following M-step which yields the updates of the model parameters,

$$ \pi_c^{(k+1)} = \frac{1}{N} \sum_{i=1}^{N}{\tau_{ic}\left(ce_i, \mathcal{G}; \Psi_i^{(k)} \right)}, $$

and

$$ \begin{array}{lll} \left( \theta_1^{(k+1)}, \ldots,\theta_C^{(k+1)} , \xi_1^{(k+1)},\ldots, \xi_N^{(k+1)} \right) &=&{\operatornamewithlimits{arg\,max}}_{\theta_1,\ldots,\theta_C, \xi_1,\ldots, \xi_N} \sum_{i=1}^{N}{\sum_{m=1}^{C}{\tau_{im}\left(ce_i, \mathcal{G}; \Psi_i^{(k)} \right) }}\\ &&{{\times\,\ln f\left(ce_i, \mathcal{G}; \theta_m^{(k)}, \xi_i^{(k)} \right)}}. \end{array} $$

As Dempster et al. (1977) show, the likelihood never decreases from one iteration to the next, i.e. \(L \left(\Psi^{(k+1)}; ce, \mathcal{G} \right) \geq L \left(\Psi^{(k)}; ce, \mathcal{G} \right)\), which makes the EM algorithm converge monotonically towards the nearest maximum of the likelihood function regardless whether this maximum is global or just local. Therefore, one may apply a stochastic extension, the Simulated Annealing Expectation Maximization (SAEM) algorithm proposed by Celeux et al. (1996), in order to overcome the EM algorithm’s tendency to converge towards local maxima. In each iteration, there is a non-zero probability that the SAEM algorithm leaves the current optimization path and starts over in a different region of the likelihood function, which results in much higher chances of finding the global maximum. But this robustness against multimodality of the objective function comes at the cost of much higher computational demands.

As the EM algorithm is computationally highly demanding, even in its basic form, and tends to become tediously slow when close to convergence, our estimation routine relies on a hybrid estimation algorithm (Render and Walker 1984): It first uses either the EM or the SAEM algorithm and takes advantage of their robustness before it switches to the direct maximization of the log likelihood by the much faster BFGS algorithm.Footnote 19 The estimation routine in this form turned out to be efficient and robust as it reliably converged towards the same maximum likelihood estimates regardless of the randomly chosen start values.

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Fehr-Duda, H., Bruhin, A., Epper, T. et al. Rationality on the rise: Why relative risk aversion increases with stake size. J Risk Uncertain 40, 147–180 (2010). https://doi.org/10.1007/s11166-010-9090-0

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