Abstract
When it comes to experiments with multiple-round decisions under risk, the current payoff mechanisms are incentive compatible with either outcome weighting theories or probability weighting theories, but not both. In this paper, I introduce a new payoff mechanism, the Accumulative Best Choice (“ABC”) mechanism that is incentive compatible for all rational risk preferences. I also identify three necessary and sufficient conditions for a payoff mechanism to be incentive compatible for all models of decision under risk with complete and transitive preferences. I show that ABC is the unique incentive compatible mechanism for rational risk preferences in a multiple-task setting. In addition, I test empirical validity of the ABC mechanism in the lab. The results from both a choice pattern experiment and a preference (structural) estimation experiment show that individual choices under the ABC mechanism are statistically not different from those observed with the one-round task experimental design. The ABC mechanism supports unbiased elicitation of both outcome and probability transformations as well as testing alternative decision models that do or do not include the independence axiom.
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Notes
Such mechanism has been called differently in previous literature, such as Random Problem Selection (RPS) in Beattie and Loomes (1997) and Azrieli et al. (2018, 2020), Random Lottery Incentive Mechanism (RLIM) in Harrison and Swarthout (2014), Random Incentive System (RIS) in Baillon et al. (2014) and Pay One Randomly (POR) in Cox et al. (2015).
RLIS is incentive compatible with EUT, and another payoff mechanism “Pay All Correlated”, proposed by Cox et al. (2015), is incentive compatible with DT. However, no currently known payoff mechanism is incentive compatible with RDU or CPT with the fixed reference point in a multiple-round experiment. The popular theories mentioned here, EUT, DT, RDU, CPT, also satisfy first-order stochastic dominance. However, the ABC mechanism does not require further assumptions on first-order stochastic dominance, just rational preferences.
There exists no incentive compatible mechanism for testing these models other than the one-round task “mechanism” in which each subject makes only one decision and all tests use only between-subjects data.
Truthful revelation or revealing truthfully henceforth means revealing single-task preference.
Take a 2-round task as an example: \({\mathscr {N}}=\{1, 2\}\) and \(2^{\mathscr {N}}=\{\emptyset , \{1\}, \{2\}, \{1, 2\}\}\) represents \(\{\)Neither round is paid, Only Round 1 is paid, Only Round 2 is paid, Both Round 1 and 2 are paid\(\}\). A probability measure \({\mathbb {P}}_1\): \({\mathbb {P}}_1(\emptyset )={\mathbb {P}}_1(\{2\})={\mathbb {P}}_1(\{1, 2\})=0,{\mathbb {P}}_1(\{1\})=1\) shows Round 1 is paid for sure. Another probability measure \({\mathbb {P}}_2: {\mathbb {P}}_2(\{1, 2\})=1,{\mathbb {P}}_2(\emptyset )={\mathbb {P}}_2(\{1\})={\mathbb {P}}_2(\{2\})=0\) represents that both Rounds 1 and 2 are paid with certainty. A third probability measure \({\mathbb {P}}_3\): \({\mathbb {P}}_3(\{1\})={\mathbb {P}}_3(\{2\})=0.5,{\mathbb {P}}_3(\emptyset )={\mathbb {P}}_3(\{1, 2\})=0\) means that either Round 1 or Round 2 is paid with probability of 0.5 each.
In Azrieli et al. (2018), the payoff mechanism \(\phi\) is defined as a mapping from subjects’ choices to the possible payments they receive and \((D,\phi )\) is referred to as a general experiment. \(\phi\) there corresponds to \({\mathbb {P}}\) here but defined from different angles to represent what to pay to the subjects. Also, in their setting, \(D=(D_1,\ldots ,D_n)\) is exogenously given and they didn’t mention I. In this paper, both D and I are parts of the payoff mechanism. In the discussion below, the task structure D and information structure I play an important role in order to achieve incentive compatibility.
Subjects are informed of D refers that they know the internal connection between their choices and the tasks if there is any including the number of rounds, n, but not the specific lotteries in each \(D_i\). Whether and when they know about the specific lotteries in each \(D_i\) is given by the information structure I.
Relate this definition with related works by Cox et al. (2015) and Azrieli et al. (2018), both of them use notations of another preference relation \(\succeq ^m\) or \(\succeq ^*\) to refer to subject’s preference given the payoff mechanism. Here, since I defined that the preference is over \({\mathscr {L}}\) consisting of all the simple and compound lotteries, I can use the symbol of the original preference directly. In essence, they all are equivalent. Cox et al. (2015) also distinguish strong and weak incentive compatibility; while in this paper, all incentive compatibility refers to their weak incentive compatibility.
Lottery A first-order stochastically dominates lottery B (denoted as \(A\ge ^{FOSD}B\)) if for all outcome x, the cumulative distribution functions of the two satisfy \(F_A(x)\le F_B(x)\). A preference relation, \(\succeq\), satisfies FOSD if \(A\succeq B\) for all A, B such that \(A\ge ^{FOSD} B\) (the strict version is: if \(A>^{FOSD}B\), then \(A\succ B\)).
In addition, in this example, even if the subject is informed of E before making any decisions, the payoff mechanism still meets all the properties in Proposition 1 and therefore is incentive compatible (since E is dominated by all other lotteries and would never be chosen by any first-order stochastic dominance preference).
T test and Pearson test provide the same qualitative results.
Together, all 47 subjects violated first-order stochastic dominance only 18 times (2.5%).
To check 2n lotteries, ABC needs 2n-1 rounds with one option from the preceding round carrying over to the following round.
The design is a simplified version similar to the experiments in Harbaugh et al. (2001). In the study, the subject pool was 7- and 11-year-old kids. In order to adapt to children’s cognitive level, the authors offer bundles of juices and chips to let them choose. In each task, all options are from the same budget line.
Even though at Round 6, we bring back one option subject discarded in Round 1, we still keep the whole lottery set fixed throughout the experiment. Since Round 6 is the last round, it is not invasive in terms of changing subjects’ anticipations about the future options.
This is not the unique implementation with ABC. An alternative path is provided in the appendices.
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Acknowledgements
I am extremely grateful to my dissertation committee (James C. Cox, Yongsheng Xu, Vjollca Sadiraj and Ajay Subramanian) for advisement and to James C. Cox, Vjollca Sadiraj and Ulrich Schmidt, Glenn W. Harrison and J. Todd Swarthout for generously sharing their data and codes. I truly appreciate the feedback and insights from the editor, the anonymous referee and Song Dai. I’m thankful to the feedback from 2016 ESA North American Conference, 2016 CEAR/MRIC Behavioral Insurance Workshop at Georgia State University and Experimental Brown Bag at University of Pittsburgh.
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Li, Y. The ABC mechanism: an incentive compatible payoff mechanism for elicitation of outcome and probability transformations. Exp Econ 24, 1019–1046 (2021). https://doi.org/10.1007/s10683-020-09688-2
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DOI: https://doi.org/10.1007/s10683-020-09688-2