Abstract
We present a simple model where preferences with complexity aversion, rather than ambiguity aversion, resolve the Ellsberg paradox. We test our theory using laboratory experiments where subjects choose among lotteries that “range” from a simple risky lottery, through risky but more complex lotteries, to one similar to Ellsberg’s ambiguity urn. Our model ranks lotteries according to their complexity and makes different—at times contrasting—predictions than most models of ambiguity in response to manipulations of prizes. The results support that complexity aversion preferences play an important and separate role from beliefs with ambiguity aversion in explaining behavior under uncertainty.
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Notes
Modeling complexity aversion as taxing utilities due to the extra (cognitive) costly effort to compute reduced-form equivalents would result in a circular argument. The reason is that, once such effort is already taken, it becomes sunk and should not affect choices. However, we do not assume that DMs reduce the lotteries, and our simple model and treatment manipulations enable us to distinguish complexity from uncertainty aversion. See below.
We model the DM as having a particular prior on the Ellsberg urn lottery only to illustrate our claim: There do exist priors with complexity aversion that resolve the Ellsberg paradox.
In contrast, the urns with n ∈ (50,100) can be viewed as twice compound and thus more complex lotteries.
We use the linear, additive structure for expositional simplicity. However, qualitative results will not change with any utility that is declining in δ.
We use one half for expositional simplicity. The arguments extend for different probability levels.
Note that δ in fact measures the difference in complexity between A and B, rather than being an absolute measure of complexity of lottery B.
Obviously, the complexity of the lottery itself may shape this distribution through the variance of the estimates in the population. This, however, does not affect any of the predictions below.
See Appendix A for more details.
Ambiguity aversion requires the concavity of ϕ, whereas complexity aversion is modeled as a linear transformation of payoffs which depends on the complexity of alternative prospects. This model is related to source-dependent approaches to uncertainty (Ergin and Gul 2009; Grant et al. 2009). See Appendix A for more details.
We prefer this more conservative prediction. However, if most of the subjects in the experiment are averse to ambiguity (concave ϕ) fewer people should prefer B over A after our prize manipulation. See Online Appendix A for more details.
To see this, note that the area A in Fig. 2b does not exist and these theories only suffer the loss of the B-preferring individuals corresponding to area B.
Appendix A.3 contains the formal analysis of these arguments.
The experimental instructions can be found in the Online Appendix.
Note that we did not multiply the low prize in the experiment. In Appendix A, we show formally that it neither affects the derived theoretical predictions of complexity aversion nor alters the predictions of the ambiguity literature.
Naturally, the terms risky, ambiguous, and compound were never used in the actual experiment.
As mentioned, we use \(\frac {1}{2}\) as the probability of high prize to simplify the exposition of complexity aversion, whereas in the actual experiments the probabilities are \(\frac {3}{8}\) in Wave 1 and \(\frac {3}{4}\) in Wave 2. The reason is to prevent subjects from focusing on the symmetric \( \frac {1}{2}\) while evaluating or estimating probabilities of prizes during the experiment. The theoretical predictions are robust to considering \(\frac { 1}{2}\), \(\frac {3}{8}\) or \(\frac {3}{4}\).
We thank the participants in the DT conference in HCE, Paris, 2009 for suggesting this design.
The two exceptions, both in Wave 2, are still close to 50%.
Wilcoxon signed-rank test rejects that the percentages are significantly different from zero in all cases (p = 0 in all cases).
Pairwise Wilcoxon signed-rank tests confirm this finding with some exceptions. First, the first percentage in each row is always lower than the third percentage (e.g. A over C vs. A over B in the first row etc.) in both tables (p < 0.002 in all cases but Wave 1 L = 50 with p < 0.045). The comparison between the first and second percentage is almost always significant at 5%, except the A row (p > 0.46 in Wave 1 and p > 0.11 in Wave 2), the D row in Wave 1 for H = 100 (p = 0.132), and the C row for H = 50 (p = 0.063). The last comparison, the second vs. third percentage, is again almost always significant at 5%, the exceptions being row A for H = 50 (p = 0.098), row C for (p = 0.065,0.698 for H = 50,100, resp.), and row B (p = 0.99,0.114 for low and high stakes, resp.). Note that most of these exceptions are at least marginally (p≤0.1) or close to marginally significant.
Since the dependent variable takes values −1, 0, and 1 if, respectively, the individual prefers the more complex lottery, she is indifferent between them, or she prefers the simpler option, we estimate the ordered-logit model. The results are generally robust to alternative specifications of the dependent variable, different estimation techniques, or panel data regressions, in which subjects’ different choices play the role of the time-series variable.
In his experiment, Halevy (2007) provides a robustness test of his main treatment, in which he multiplies the stakes by 10. We used his data and found that 22.2% prefer the Ellsberg-like lottery to the risky one for low stakes, but this fraction increases to 25% if stakes are higher. The difference is not statistically significant, though. There are three reasons why this additional finding may support complexity aversion: (i) any increase already contradicts most of the theories of ambiguity aversion, (ii) there are only 38 observations in the high-stake treatment, (iii) multiplying the stakes by 10 should increase subjects’ estimation effort considerably and thus work against the prediction of complexity aversion. In contrast, his ten-ball lotteries are more complex than ours. Hence, we consider his data stimulating for further exploration of complexity aversion but prefer to be cautious using this data to make general conclusions concerning our theory.
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Our paper greatly benefited from comments and suggestions of a referee and the Editor. We also thank Itzhak Gilboa, Yoram Halevy, Brian Hill, Dimitry Mezhvinsky, Jim Peck, and participants of the DT conference 2009 in HCE Paris and SEET 2013. Jaromír Kovářík greatly acknowledges support from Spanish Ministry of Science and Innovation (ECO 2012-31626, ECO 2012-35820), the Basque Government (IT-783-13), and GACR (14-22044S).
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Kovářík, J., Levin, D. & Wang, T. Ellsberg paradox: Ambiguity and complexity aversions compared. J Risk Uncertain 52, 47–64 (2016). https://doi.org/10.1007/s11166-016-9232-0
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DOI: https://doi.org/10.1007/s11166-016-9232-0