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Detecting heterogeneous risk attitudes with mixed gambles

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Abstract

We propose a task for eliciting attitudes toward risk that is close to real-world risky decisions which typically involve gains and losses. The task consists of accepting or rejecting gambles that provide a gain with probability \(p\) and a loss with probability \(1-p\). We employ finite mixture models to uncover heterogeneity in risk preferences and find that (i) behavior is heterogeneous, with one half of the subjects behaving as expected utility maximizers, (ii) for the others, reference-dependent models perform better than those where subjects derive utility from final outcomes, (iii) models with sign-dependent decision weights perform better than those without, and (iv) there is no evidence for loss aversion. The procedure is sufficiently simple so that it can be easily used in field or lab experiments where risk elicitation is not the main experiment.

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Notes

  1. Harrison and Rutström (2009) apply finite mixture models too in order to distinguish EU from non-EU behavior. However, they classify decisions instead of subjects. Therefore, their results are not directly comparable to Bruhin et al. (2010) and Conte et al. (2011).

  2. Mixed gambles bring the experimental task closer to real-world risky decisions than most previous experimental work. Real-world decisions typically involve gain–loss gambles, but most prior experimental research on the elicitation of risk attitudes used lotteries involving only gains (e.g., Brunner et al. 2007; Ebert and Wiesen 2009; Deck and Schlesinger 2010) or lotteries involving either only gains or only losses (e.g., Holt and Laury 2002; Fehr-Duda et al. 2010; Bruhin et al. 2010).

  3. Most of prior experimental work on reference-dependent models has been limited to testing a subset of the behavioral features of PT. For example, Abdellaoui (2000) and Bruhin et al. (2010) use lotteries involving either only gains or only losses and estimate probability distortion parameters for the gain and loss domains but do not estimate a loss aversion parameter since that is neither feasible nor meaningful. Tversky and Kahneman (1992) and Abdellaoui et al. (2007) study all features of risk preferences proposed by PT but use hypothetical choices. Abdellaoui et al. (2008) use real incentives for lotteries with only gains but not for lotteries with only losses or with gains and losses. As far as we know, there are only two other studies that analyze all aspects of PT using monetary incentives for all types of lotteries: Harrison and Rutström (2009) and Tanaka et al. (2010).

  4. The regressions reported in Table 5 exclude the eight multiple switchers so that the number of accepted lotteries corresponds to the switching point. The subsequent analysis includes all 109 subjects.

  5. Given that our gambles only have three probability values and that we will measure loss aversion in the PT model, we cannot identify a two-parameter probability weighting function.

  6. The function \(u\) has a kink at \(r\) and is smooth everywhere else. The kink is caused by loss aversion, and does not reflect an intrinsic value of outcomes. That is, it is plausible that the basic utility function is differentiable at \(r\). Thus, they define the loss aversion index as \(\lambda =u_{\uparrow }^{\prime }(x|r)/u_{\downarrow }^{\prime }(x|r),\) where \(u_{\uparrow }^{\prime }(x|r)\) denotes the left, and \(u_{\downarrow }^{\prime }(x|r)\) the right derivative of \(u\) at \(r\) (both derivatives are assumed to exist and be positive and finite). If the loss aversion index is equal to 1, there is no loss aversion. If the index is greater than 1, the agent is loss averse.

  7. Note that the above individual likelihood contribution is highly non-linear even after taking logs. Maximizing the finite mixture model’s likelihood is therefore not trivial and standard numerical maximization techniques, such as the BFGS algorithm for example, will usually fail in finding its global maximum. We therefore apply Dempster et al’s (1977) expectation maximization (EM) algorithm to obtain the model’s maximum likelihood estimates \(\hat{\Psi }\). Instead of maximizing the finite mixture model’s complete log likelihood, the EM algorithm proceeds iteratively in two steps: In the E-step, it computes the individual ex-post probabilities of type-membership given the actual fit of the model. In the subsequent M-step, the EM algorithm updates the model’s fit using these ex-post probabilities to maximize each type’s log likelihood contribution separately.

  8. Lo et al. (2001) proposed a statistical test (LMR-test) to select among finite mixture models with varying numbers of types, which is based on Vuong’s (1989) test for non-nested models. However, the LMR-test is unlikely to be suitable when the alternative is a single-type model with strongly non-normal outcomes (Muthén 2003).

  9. The finding is also consistent with evidence from field studies. For example, both Snowberg and Wolfers (2010) and Jullien and Salanié (2000) find that inverse s-shape probability weighting dominates in explaining why there is overbetting on the long-shot horse while favorites are underbet. In the context of insurance choices, Cohen and Einav (2007) and Barseghyan et al. (2010) find preferences in which inverse s-shape probability weighting plays the most important role.

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Acknowledgments

We thank James Cox, Erik Eyster, Glenn Harrison, Botond Koszegi, and Peter Wakker for various comments and suggestions. We also thank seminar and conference participants at HEC Paris, University of Lausanne, University Jaume I Castellon, Economic Science Association World Meeting in Copenhagen, International Meeting on Behavioral and Experimental Economics in Barcelona, D-TEA Workshop in Paris, and the 68th European Meetings of the Econometric Society in Toulouse. Support from HEC Paris Foundation, Swiss National Science Foundation grant 135602, NOVA FORUM, and Fundação para a Ciência e Tecnologia is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Faculty of Business and Economics (HEC Lausanne), University of Lausanne, Internef 535, 1015, Lausanne, Switzerland

    Luís Santos-Pinto & Adrian Bruhin

  2. Nova School of Business and Economics, INOVA, Universidade Nova de Lisboa, Lisbon, Portugal

    José Mata

  3. HEC-Paris, Jouy-en-Josas, France

    Thomas Åstebro

Authors
  1. Luís Santos-Pinto
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  2. Adrian Bruhin
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  3. José Mata
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  4. Thomas Åstebro
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Corresponding author

Correspondence to Luís Santos-Pinto.

Appendices

Appendix

Table 9 reports the parameter estimates of the mixture models where the non-EU have RDU or WU preferences, respectively.

Table 9 Parameter estimates of non-reference-dependent mixture models
Full size table

Figure 2 displays the probabilities of EU-type-membership for the two models. We see that both models cleanly segregate subjects into the two types. In the model where the non-EU types have WU preferences, we find 58 EU subjects and 51 non-EU subjects. In the model where the non-EU types have RDU preferences, we find 55 EU subjects and 54 non-EU subjects. In contrast to the reference-dependent models, however, the individual classifications of the non-reference-dependent models differ considerably: the RDU and the WU models only classify 54 out of the 109 subjects into the same types. Table 10 reports likelihoods for the BIC and AIC information criteria.

Fig. 2
figure 2

Distribution of individual probabilites of EU-type-membership

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Table 10 BIC and AIC
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Experimental material: detecting heterogeneous risk attitudes with mixed gambles

Research Study.

Decision Making under Risk.

Instructions.

Read these instructions carefully as your understanding of them will affect your ability to earn money.

Your decision sheet shows a table with ten decisions listed on the left. Each decision is a choice between “Accept” and “Reject” the lottery. If you Accept the lottery you have the chance of either winning or losing some money. If you Reject the lottery, you will not earn anything and you will not lose anything—your payoff will be zero.

You will make ten choices and record these by ten check marks, but only one of them will be used in the end to determine your earnings. Before you start making your ten choices, please let me explain how these choices will affect your earnings for this part of the experiment.

We will use a ten-sided die to determine payoffs; the faces are numbered from 1 to 10 (the “0” face of the die will serve as 10.) After you have made all of your choices, we will throw this die twice, once to select one of the ten decisions to be used, and a second time to determine your payoff if you chose “Accept” for that lottery.

Even though you will make ten decisions, only one of these will end up affecting your earnings, but you will not know in advance which decision will be used. Obviously, each decision has an equal chance of being used in the end.

Now, please look at Decision 1 at the top of Example Problem Set on the next page (Table 11). If you Accept the lottery, you will earn 50.00 CHF if the throw of the ten-sided die is 1, and you will lose 5.00 CHF if the throw is 2–10. The other Decisions are similar, except that as you move down the table, the value of the highest payoff decreases. For each Decision, you are asked to choose whether or not you want to take the gamble by checking the Accept or Reject columns.

This table is just an example and is not to be used to make real decisions. The problems for which you will be asked to make decisions have a similar structure, but the magnitudes of the gains and losses will be different, as well as the numbers on the dies that will determine if you make a gain or a loss.

Your gains—You may win or lose money if you Accept the lottery. If you win, you will be paid the amount won; if you lose you will have to pay the amount lost. The maximum amount that you can lose in the gambles is CHF 65. However, because we do not want you to end with less money than you started out you will be given CHF 65 from which we will deduct any loss that you may experience. In addition, you will be paid an additional CHF 15 as a compensation for your time spent.

Table 11 Example problem set (not to be used for actual decisions)
Full size table

There will be a total of three problem sets (A B, and C), each composed of ten decisions. In each problem set, the gains, losses and chances to win will be different. In the end, we will only use one of the three problem sets to determine your earnings. The choice of which problem set to use to determine earnings will be made using a six-sided die. Each problem set has the same probability of being chosen. If the outcome of the six-sided die is either 1 or 2, problem set A counts, if the outcome is either 3 or 4 set B counts, and if the outcome is either 5 or 6 set C counts.

To summarize, in each problem set, you will make ten choices: for each choice, you will have to decide between Accept or Reject the lottery. You may choose Accept for some decisions and Reject for other decisions, and you may change your decisions and make them in any order. When you are finished with making all ten choices, you will move on to the next problem set. This will be repeated three times. After the three problem sets are done, we will ask you a few demographic questions about yourself.

At the end, a randomly chosen participant will first cast a six- sided die, numbered from 1 to 6, and then a 10-sided die, numbered 0 to 9 to select the decision which determines your earnings. The 6-sided die will decide which problem will be used, and the 10-sided die will decide which decision will be used to determine your gains or losses. The same participant will then throw the 10-sided die again to choose whether you win or lose on that lottery if you decided to Accept that lottery. You will be paid all earnings in cash when finished.

Are there any questions? Now you may begin making your choices. Please, do not talk with anyone while we are doing this; raise your hand if you have a question.

Problem set A

1.1 Instructions

Your decision sheet shows ten decisions listed on the left (Table 12). Each decision is a choice between “Accept” and “Reject” the lottery. If you Accept the lottery, you have the chance of either winning or losing some money. If you Reject the lottery, you will not earn anything and you will not lose anything—your payoff will be zero.

You will make ten choices and record these by ten check marks, but only one of them will be used in the end to determine your earnings.

In this problem set, the gains and losses associated to each choice is different from before. Also, note that whether there will be gains or losses may be determined by different numbers on the dice. Once you have completed your choices you will move on to the next problem set. Earnings will be determined at the end of the study.

Table 12 Problem set A
Full size table

Please check whether you Accept or Reject each lottery.

Problem set B

1.1 Instructions

As before, your decision sheet shows ten decisions listed on the left (Table 13). Each decision is a choice between “Accept” and “Reject” the lottery. If you Accept the lottery, you have the chance of either winning or losing some money. If you Reject the lottery you will not earn anything and you will not lose anything—your payoff will be zero.

You will make ten choices and record these by ten check marks, but only one of them will be used in the end to determine your earnings.

In this problem set, the gains and losses associated to each choice is different from before. Also, note that whether there will be gains or losses may be determined by different numbers on the dice. Once you have completed your choices you will move on to the next problem set. Earnings will be determined at the end of the study.

Please check whether you Accept or Reject each lottery.

Table 13 Problem set B
Full size table

Problem set C

1.1 Instructions

As before, your decision sheet shows ten decisions listed on the left (Table 14). Each decision is a choice between “Accept” and “Reject” the lottery. If you Accept the lottery, you have the chance of either winning or losing some money. If you Reject the lottery, you will not earn anything and you will not lose anything—your payoff will be zero.

You will make ten choices and record these by ten check marks, but only one of them will be used in the end to determine your earnings.

In this problem set, the gains and losses associated to each choice is different from before. Also, note that whether there will be gains or losses may be determined by different numbers on the dice. Once you have completed your choices you will move on to the next problem set. Earnings will be determined at the end of the study.

Please check whether you Accept or Reject each lottery.

Table 14 Problem set C
Full size table

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Santos-Pinto, L., Bruhin, A., Mata, J. et al. Detecting heterogeneous risk attitudes with mixed gambles. Theory Decis 79, 573–600 (2015). https://doi.org/10.1007/s11238-015-9484-1

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