Stake effects on ambiguity attitudes for gains and losses

Bouchouicha, Ranoua; Martinsson, Peter; Medhin, Haileselassie; Vieider, Ferdinand M. Article — Published Version Stake effects on ambiguity attitudes for gains and losses Theory and decision: an international journal for multidisciplinary advances in decision science Provided in Cooperation with: WZB Berlin Social Science Center Suggested Citation: Bouchouicha, Ranoua; Martinsson, Peter; Medhin, Haileselassie; Vieider, Ferdinand M. (2017) : Stake effects on ambiguity attitudes for gains and losses, Theory and decision: an international journal for multidisciplinary advances in decision science, ISSN 1573-7187, Springer US, New York, Vol. 83, Iss. 1, pp. 19–35-, https://doi.org/10.1007/s11238-016-9585-5 This Version is available at: https://hdl.handle.net/10419/167628 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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(2017) 83:19–35 DOI 10.1007/s11238-016-9585-5 Stake effects on ambiguity attitudes for gains and losses Ranoua Bouchouicha1·Peter Martinsson2,3· Haileselassie Medhin2,4·Ferdinand M. Vieider5,6 Published online: 5 January 2017 © The Author(s) 2017. This article is published with open access at Springerlink.com Abstract We test the effect of stake size on ambiguity attitudes. Compared to a baseline condition, we find subjects to be more ambiguity seeking for small-probability gains and large-probability losses under high stakes. They are also more ambiguity averse for large-probability gains and small-probability losses. We trace these effects back to stake effects on decisions under risk (known probabilities) and uncertainty (unknown probabilities). For risk, we replicate previous findings. For uncertainty, we find an increase in probabilistic insensitivity under high stakes that is driven by increased uncertainty aversion for large-probability gains and for small-probability losses. Keywords Monetary stakes ·Ambiguity aversion ·Risk attitudes Financial support from the University of Munich and the Swedish Research Council is gratefully acknowledged. We are grateful for the support given us by the Ethiopian Development Research Institute. BFerdinand M. Vieider [email protected] 1Henley School of Business, University of Reading, Reading, UK 2Department of Economics, University of Gothenburg, Gothenburg, Sweden 3Linköping University, Linköping, Sweden 4Ethiopian Development Research Institute, Addis Ababa, Ethiopia 5Department of Economics, University of Reading, Reading, UK 6WZB Berlin Social Science Center, Berlin, Germany 123 20 R. Bouchouicha et al. 1 Introduction Some form of uncertainty is involved in most important decision-making processes, and uncertainty about the future may, indeed, be considered inherent to the human condition. It is thus not surprising that how people react to such uncertainties has received considerable attention in decision theory. Most empirical investigations have, however, concentrated on the special case of risk, where objective probabilities are known. While this case may be a good representation of some decision situations, such as roulette wheels or many medical decision problems (for which precise probabilities can be derived from a large number of past observations), the case of uncertainty proper, for which no objectively derivable probabilities are available, is likely to be much more important in praxis. Notwithstanding its importance in the real world, such uncertainty has until recently mostly been studied in opposition to behavior under risk, and even so mostly for a restricted class of 50-50 gambles over gains (see Trautmann and van de Kuilen (2015), for an overview). The phenomenon by which people tend to prefer outcome generating processes characterized by known probabilities (risk) over outcome generating processes with vague or unknown probabilities (uncertainty) which are normatively equivalent under subjective expected utility theory (Savage 1954) is known as ambiguity aversion (Ellsberg 1961). In addition to the challenges it creates for subjective expected utility, ambiguity aversion has been deemed to be an important determinant of a variety of real-world decisions, including the home bias (Kilka and Weber 2000), reluctance to vaccinate (Ritov and Baron 1990), and the preference for established brands over new ones (Muthukrishnan et al. 2009), to name but a few. Only recently has a more systematic way of measuring ambiguity attitudes been developed, which allows for measurements over the whole probability spectrum and for less artificial types of uncertainty (Abdellaoui et al. 2011,2005;Fox and Tversky 1998;Kilka and Weber 2000;Maafi 2011;van de Kuilen and Wakker 2011). Given the recency of these developments, many issues remain to be explored. This paper specifically investigates whether ambiguity attitudes are affected by the magnitude of the monetary outcomes at stake, a problem on which no evidence exists to date. Ambiguity attitudes are thereby estimated for different stake levels over the whole probability space. In addition, we can trace any stake effects we find for ambiguity back to its constituent parts—stake effects for risk and stake effect for uncertainty— and thus determine what underlying patterns drive stake effects for ambiguity. Contrary to most of the literature on stake effects under risk, we use a between-subjects design which avoids potential contrast effects resulting from the direct contraposition of high and low stake decisions (Greenwald 1978;Read 2005). The issue of stake effects on ambiguity and uncertainty attitudes is important to the extent that real-world decisions involving more or less well-known probabilities may concern a diverse spectrum of outcomes, ranging from relatively trivial stakes to extremely high ones. Any differences in ambiguity attitudes dependent on stakes can thus be informative about this type of behavior. Furthermore, possible stake effects under uncertainty provide information on the extent to which results obtained with low stakes in the laboratory can be generalized to decisions with non-trivial stakes in the field, or on how any such generalizations might need to be adjusted for differences 123 Stake effects on ambiguity attitudes for gains and losses 21 in stakes. Since we elicit certainty equivalents (CEs) for a number of prospects with different probabilities, we can analyze how such a stake effect manifests itself over the whole probability space. Furthermore, we present evidence for both gains and losses. This may be interesting for analyzing decisions, e.g., on the stock market, which can be characterized as decisions under true uncertainty, and which have been linked to ambiguity aversion In contrast to uncertainty, the effect of different stake sizes has been studied extensively for the case of risk, especially for gains. In an experiment conducted in rural India, Binswanger (1980) showed that risk aversion increases in stake levels. Kachelmeier and Shehata (1992) ran high stakes experiments in China, finding that risk seeking for small probabilities was reduced relative to smaller stakes, a result that was replicated by Lefebvre et al. (2010) in France using a between subjects design. Holt and Laury (2002) supplied further evidence on how subjects become more risk averse as stakes increase, and Fehr-Duda et al. (2010) showed with Chinese subjects how, in a prospect theory framework, such stake effects will affect the probability weighting function as well as the utility function, thus violating the separability precept by which attitudes towards money are supposed to be reflected purely in utility (although this may, to some extent, depend on the functional form of utility—see Scholten and Read (2014), and Bouchouicha and Vieider (2016)). Stake effects are less well understood for risky losses. Hogarth and Einhorn (1990) found no effect of such stake variations on losses. Etchart-Vincent (2004) tested different hypothetical stake levels against each other, and found only slight evidence for increased risk aversion under high stakes. Bosch-Domènech and Silvestre (2006), on the other hand, found large increases in risk aversion for both small and large probabilities of losses as stakes increased. Fehr-Duda et al. (2010) found inconclusive effects in the loss domain. Moreover, there exist several papers investigating the issue of whether hypothetical choice differs from choices for real money under losses (Etchart-Vincent and L’Haridon 2011;Laury et al. 2009;Schoemaker 1990;Vieider 2011), which is, however, quite a different issue. We provide some additional evidence on the effect of stake sizes for decisions under risk in the loss domain. We conducted an experiment in Ethiopia where we compare a baseline condition, providing already relatively high stakes, to a high stake condition in which all monetary amounts are doubled in a clean between subjects design. For gains, high stakes cause increases in ambiguity seeking for small probabilities and increases in ambiguity aversion for large probabilities relative to the baseline. This means that under high stakes, we observe more extreme jumps at the endpoints of the probability scale, resulting in reduced sensitivity to probabilistic change in the intermediate ranges. We next trace this effect back to stake effects for risk and uncertainty. For known probabilities, we replicate the well-known finding of risk aversion increasing in stake size—an effect that is uniform over the probability space. For unknown probabilities, we find uncertainty aversion to increase in stake sizes just as for risk. This effect, however, happens entirely through increased uncertainty aversion for large probabilities with uncertainty attitudes for small probabilities remaining unchanged, thus resulting in reduced probabilistic sensitivity for intermediate probabilities. The effect found under ambiguity is thus driven mainly by stake effects in decisions under uncertainty. This pattern is mirrored for losses. In the baseline condition, we find a pattern of ambiguity 123 22 R. Bouchouicha et al. aversion for low probability levels and ambiguity seeking for high probability levels. Under high stakes, we again find an accentuation of these patterns at the endpoints of the probability scales, with subjects being more ambiguity averse for small probabilities and more ambiguity seeking for large probabilities. Once again, this pattern is driven by more extreme uncertainty attitudes at the endpoints of the probability scale, while we find no effect of stakes on risk attitudes. 2 Experiment: setup and method Subjects. A total of 157 subjects were recruited by the authors at Addis Ababa University, in Ethiopia, on the campus of Economics, Management, and Information Sciences. The experiment was conducted in English—the teaching language of the university—and one of the authors, who conducted the experiments and is fluent in Amharic (the local language), provided additional explanations where needed. A total of 12 subjects had to be excluded, because they manifestly did not understand the task or did not respond to all questions. The average age was 21.1 years, and 73% of subjects were male. Tasks. We elicited certainty equivalents (CEs) for simple binary prospects through choice lists. This procedure involved making subjects choose between a prospect and different sure amounts of money increasing in steps of 3 Birr from the low amount in the prospect to the high amount of the prospect, with the amounts being mirrored for losses, which took place from an endowment. The tasks are shown in Table 1, and the complete instructions can be found in the online appendix or on www.ferdinandvieider. com/instructions.html. Subjects were told that there were no right or wrong answers and that we were only interested in their preferences. They were also told, however, that we were interested in the point where they would switch from preferring the prospect to preferring the sure amount (the opposite for losses), and that they should not switch back and forth between prospect and sure amount. For unknown probabilities, the elicitation of CEs has the advantage of producing a measure of the strength of preferences, while avoiding biases that result from the elicitation of willingness to Table 1 Decision tasks, amounts in Ethiopian Birr Baseline High stakes Gains Losses Gains Losses (1/8: 120; 0) (1/8: −120; 0) (1/8: 240; 0) (1/8: −20; 0) (1/8: 120; 30) (1/8: −120; −30) (1/8: 240; 60) (1/8: −240; −60) (2/8: 120; 0) (2/8: −120; 0) (2/8: 240; 0) (2/8: −240; 0) (3/8: 120; 0) (3/8: −120; 0) (3/8: 240; 0) (3/8: −240; 0) (5/8: 120; 0) (5/8: −120; 0) (5/8: 240; 0) (5/8: −240; 0) (6/8: 120; 0) (6/8: −120; 0) (6/8: 240; 0) (6/8: −240; 0) (7/8: 120; 0) (7/8: −120; 0) (7/8: 240; 0) (7/8: −240; 0) (7/8: 120; 30) (7/8: −120; −30) (7/8: 240; 60) (7/8: −240; −60) 123 Stake effects on ambiguity attitudes for gains and losses 23 pay or willingness to accept measures (Trautmann et al. 2011). The CEs were elicited using paper and pencil and keeping the order of the tasks constant throughout the experiment. Risky gains were followed by uncertain gains. Once this first part of the experiment was over, subjects received a second set of instructions concerning the loss part. Again, CEs for risky losses were elicited first and then CEs for uncertain losses were elicited. While such a fixed ordering of tasks may induce order effects, we felt that such effects were preferable to the increased noise levels that were likely to result from a randomization of the order in which the decision tasks were presented. Furthermore, a large-scale pilot revealed that no such order effects occurred in our design. Losses took place from an endowment. Etchart-Vincent and L’Haridon (2011) tested whether such losses from endowment would result in different effects from actual losses and found no such differences. They also found no order effect for losses being administered before or after gains. Risky prospects were represented using eight consecutively numbered balls in an urn. Uncertain prospects, implemented through Ellsberg-type urns, were also represented through eight numbered balls, and subjects were told that while there were always eight balls in the vague-probability urn just as in the risky one, it was possible that some numbers may be absent, while others would occur repeatedly (the names risky and uncertain were not used in the experiment, the urns were called transparent and opaque instead). This procedure follows closely the one used by Abdellaoui et al. (2011), except that we used numbers instead of colors to be able to print instructions in black and white. Treatments and payoffs. 83 subjects participated in the baseline condition and 62 in the high stake condition in a between-subjects design. The gender ratio did not significantly differ between treatments (z=0.336,p=0.73). While within-subject designs have often been employed for stake comparisons in the literature, we chose this alternative to avoid contrast effects that can potentially result in experimenter demand effects in within-subject designs (Greenwald 1978;Read 2005). Running the experiment in Ethiopia allowed us to offer non-trivial payoffs to the subjects. The monetary prize to be won in the baseline condition was 120 Birr. This is the PPP equivalent of e20 and it corresponds to about 20–25% of the total monthly expenditure of the median participant in the experiment. In the high stake condition, all monetary amounts were doubled. While this may be considered a relatively modest increase in stake sizes compared to other manipulations in the literature, this also means that any effects found are all the more notable. Notice also that, while some previous studies on stake sizes used higher stake differences between treatments, they typically did so from a much lower baseline.1We chose a higher baseline on purpose to determine whether choices would further change as compared to typical experimental payoffs, even when the latter are already on the high side. At the end of the experiment, one of the decisions was randomly selected for real pay—the standard procedure in this kinds of experiment. Estimation and analysis. Stake effects can be difficult to analyze within a certain decision model. For instance, prospect theory (Kahneman and Tversky 1979;Tversky and Kahneman 1992)—the prevalent descriptive theory of decision-making under risk 1For instance, Holt and Laury (2002) increased their stakes 20-fold and Lefebvre et al. (2010) by factors of four and ten. The baseline in the first study, however, was about $2, and it was about e4 in the second. 123 24 R. Bouchouicha et al. Fig. 1 Illustration of indices and uncertainty today (Starmer 2000;Wakker 2010)—represents decisions through utility and probability weighting functions, which both contribute to risk attitudes. This, however, means that any effects of stakes on risk attitudes may affect both utility and probability weighting and, in some cases, produce opposite effects on the two, such that the final effect on risk attitudes may be unclear (Vieider 2012). It is thus convenient to employ some indices that summarize the information over the probability space, so as to systematically organize the information derived from several certainty equivalents, without recurring to any particular theory to organize the results. To this end, we estimate the best linear fit for the CEs by maximum-likelihood estimation at the individual level: cei x=c+s×evi x(1) where Xis the highest outcome or prize to be won in the prospect and iis an index designating the particular prospect at hand. This regression line is estimated separately for gains and losses, and a number of indices are derived from it. The indices are illustrated in Fig. 1. A positive intercept, c, indicates a certainty equivalent for small probabilitiesthatislargerthan theexpectedvalue (EV) of the prospect in absoluteterms (indicating risk seeking for gains and risk aversion for losses), and a slope, s, smaller than 1 indicates that CEs increase less than linearly in the EV of the prospect and is thus an index of the sensitivity of CEs to changes in probability. We estimate these indices separately for risk, designated by the subscript r, and uncertainty, designated by the subscript u. Ambiguity attitudes are then quantified by the difference between the indices obtained under risk and under uncertainty, as indicated by the arrows. In addition to the intercept and the slope, we use two more indices to make the exposition more intuitive (see Abdellaoui et al. 2010, for a discussion). One is the anti-intercept, that is the distance of the regression line from 1 at p=1, which is given by 1 −s−c, and a positive value of which indicates risk aversion for gains or risk seeking for losses. Finally, we can construct a global measure of average attitudes over the probability space by subtracting the intercept from the anti-intercept, 1−s−2c 123 Stake effects on ambiguity attitudes for gains and losses 25 (Abdellaoui et al. 2011). This index takes the value of zero under risk neutrality, is positive if there is a tendency towards risk aversion (risk seeking for losses), and is negative if the general tendency goes towards risk seeking (risk aversion for losses). While risk attitudes are fully characterized by any two of these four measures, it may sometimes be convenient to present results for all four of them. Notice that how these indices are formally equivalent to the indices employed by Abdellaoui et al. (2011) to analyze neo-additive weighting functions (Chateauneuf et al. 2007). Notice also, however, that, in our case, these indices do not represent attributes of a probability weighting function, but a linear best fit for our non-parametric data. Noise and violations of stochastic dominance. Overall, noise levels in the data were rather high. Such high noise levels are not unusual for developing countries (L’Haridon and Vieider 2016). Some subjects systematically violated the first-order stochastic dominance, which resulted in a negative slope of the regression line described above. Violations of stochastic dominance are common even amongst highly educated subjects in the West (Birnbaum 1999). Since such preferences violate basic principles of rationality, we excluded subjects who had a negative slope for any of the four decision domains involved—risky gains, uncertain gains, risky losses, or uncertain losses. This brought us to eliminate 13 subjects in the high stake condition and 19 subjects in the baseline condition. The results reported below do not change qualitatively if these subjects are included in the analysis. Statistics. The indices developed above constitute a convenient tool for statistical analysis, since they can be estimated at the individual level. Graphical displays use median parameter estimates throughout. All statistics reported are Mann–Whitney tests where two different variables or indices are compared, and Wilcoxon signedrank tests where indices or variables are compared to a normative benchmark. All p-values reported are two-sided, unless otherwise specified. Hypotheses. We hypothesize that risk aversion will increase in stake size for all probability levels for gains. For losses, we hypothesize that stakes will have no effect on risk attitudes. Since there is no evidence so far on stake size effects on decisions under uncertainty, our hypotheses are by necessity more tentative. We anticipate the same qualitative effects as for risk, although quantitatively the effects may be weaker given lower levels of sensitivity under uncertainty. Finally, we attempt no direct hypotheses about effects on ambiguity attitudes, since the latter depend on the quantitative differences between the effects on risk and on uncertainty, for which we have no clear predictions. 3 Results 3.1 Ambiguity attitudes for gains We start by looking at stake effects on ambiguity attitudes, represented in Fig. 2, with the probability of winning represented on the x-axis and ambiguity aversion represented on the y-axis. Ambiguity attitudes are now represented by the area between each of the two lines and the x-axis (which can be seen as representing normalized 123 26 R. Bouchouicha et al. Fig. 2 Ambiguity aversion for gains by stake size risk attitudes, since ambiguity attitudes are defined as the difference between risk and uncertainty attitudes), with areas above the x-axis representing ambiguity aversion and areas below the x-axis representing ambiguity seeking (i.e., the distance between risk and uncertainty attitudes shown by the solid arrow in Fig. 1is now given by a negative distance from zero, and the distance captured by the dashed arrow is given by a positive distance from zero, representing ambiguity aversion for gains). In the baseline condition, we observe ambiguity aversion over the whole probability space, although we cannot exclude the null hypothesis of an intercept equal to zero (z=1.18,p=0.24), indicating no ambiguity aversion for small probabilities. There is, however, clear ambiguity aversion for large probabilities, with an anti-intercept significantly larger than zero (z=4.37,p<0.001). There is also clear ambiguity aversion at the global level, as indicated by a significantly positive index of global attitudes (z=5.18,p<0.001). While for small probabilities, we find subjects to be marginally more ambiguity seeking under high stakes relative to the baseline (z=1.73,p=0.083), subjects are considerably more ambiguity averse for large probabilities under high stakes compared to the baseline (z=3.09,p=0.002). While the opposing trends at the two endpoints cancel out at the global level, so that there is no differencein termsof globalor averageambiguity attitudes(z=1.03,p=0.31), there is a considerable change in the slope indicating a movement towards more extreme attitudes at the two endpoints of the probability spectrum (z=3.03,p=0.002).2 2Notice that how a decrease in probabilistic sensitivity is now indicated by a steeper line. This is due to the fact that the line represents sensitivity under risk minus sensitivity under uncertainty. The steeper line that we find under high stakes may thus derive either from decreased sensitivity under uncertainty keeping sensitivity for risk constant or increased sensitivity under risk keeping sensitivity under uncertainty constant (or a combination of both). This issue is discussed below. 123 Stake effects on ambiguity attitudes for gains and losses 33 5 Conclusion The main objective of this paper was to test the effect of stake size on ambiguity attitudes. We found that under high stake ambiguity, attitudes become more extreme, with increased ambiguity seeking for small-probability gains and large-probability losses relative to the baseline condition. Moreover, we also found increased ambiguity aversion for large-probability gains and small-probability losses. We then traced these effects back to stake effects under risk and uncertainty. For risk, we replicated the classic result of increased risk aversion for gains under large stakes. We also brought additional evidence indicating that such risk aversion increases uniformly over the probability space, and that for losses, stakes have no effect on decisions. For uncertainty, we found high stakes to increase global uncertainty aversion for gains and not to affect global uncertainty aversion for losses. For both uncertain gains and losses, however, we found incentives to affect the relative incidence of uncertainty aversion towards the endpoints of the probability scale. In particular, subjects were found to be more uncertainty averse for large-probability gains and small-probability losses under high stakes. Taken together, these findings may help explaining behavior in situations involving small-probability large outcome losses, such as insurance decisions and stock market participation. 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