Abstract
This paper reports an experimental comparison of attitudes toward time and toward money in experience-based decisions. Preferences were elicited under rank-dependent utility for prospects with two or three consequences expressed either in time or in monetary units. Probabilities were unknown but learned through sampling. More specifically, time and money were compared under two conditions. In a first experiment, both consequences and probabilities of prospects were unknown and learned through sequential sampling. In a second experiment, the possible consequences were revealed after the sampling. A real incentive system was implemented for both time and money. The heterogeneity of preferences was assessed for time and for money through individual and mixed modeling estimations. We observe that the nature of consequences (time or money) modifies probability weighting in terms of elevation and sensitivity. Subjects exhibit more optimism and less sensitivity to probability changes when deciding about time than about money. Revealing the consequences impacts the shape of the utility function and leaves probability weighting unchanged. We also observe that the real incentives have no effect except for the reduction in decision errors. This effect is stronger for money than for time.
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Notes
We accounted for between-subject heteroscedasticity by allowing \(\sigma \) to vary across individuals. However errors are assumed to be homoscedastic across choices. If this assumption is violated, standard errors may be biased. For this reason, inference relies either on bootstrap or on a comparison of individual-level parameters.
Halton draws were generated by the halton function provided by the randomtoolbox package, in the R software (The R development core team 2005).
The range of possible time gains also varied from \(0\) to \(1\) hour as in the present study. Regarding money, the authors considered hypothetical payoffs over [\(0\) €, \(1200\) €]. In the present study, smaller monetary consequences were considered (up to \(150\) €). This allowed for the implementing of real incentives for money as well as for time.
A case of risk was considered under EBD by Barron and Ursino (2013). Subjects could sample without replacement, which allowed them to learn the objective probabilities.
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Acknowledgments
This project was part of a Ph.D dissertation, entitled “Attitudes towards Uncertainty with Monetary and Time Consequences: Experimental Investigations”. The Ph.D program was funded by MEDDE/DRI. The paper benefited from the valuable suggestions of the Ph.D committee. The authors thank Anisa Shyti and Corina Paraschiv for their reading and helpful remarks. They are also grateful to an anonymous reviewer and the associate editor for their constructive comments.
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Appendix
Appendix
1.1 Appendix 1: Modified bisection method
The bisection method was implemented as follows. For a given prospect \(\left( x,p_x, y, p_y, z\right) \), the [\(z,x\)] interval was bisected in order to locate its certainty equivalent \(c^{*}\). The procedure began with a certain offer \(c_1\) such as \(c_1= \mathrm{EV}\left( P\right) \pm 10\). If \(c_1\) was preferred, then the certainty equivalent \(c^{*}\) stood in the \([z, c_1]\) interval. In this case, a second certain offer \(c_2\) was proposed, where \(c_2\) was the midpoint of the \([z, c_1]\) interval. Otherwise, \(c^{*}\) was in the \([c_1, x]\), and \(c_2 =c_1+ 0.5\left( x-c_1\right) \) was proposed.
The procedure was iterated until the certainty equivalent was restrained to an interval \([c_\mathrm{inf}, c_\mathrm{sup}]\), with \(c_\mathrm{sup}-c_\mathrm{inf}< 5\) (for money, \(2\) for time). While the standard bisection method would stop here, the adapted method introduced two last choices that allowed us to confirm that the certainty equivalent \(c^{*}\) was in the interval. The two last choices involving \(c_\mathrm{sup}\) and \(c_\mathrm{inf}\) were repeated. In Table 7, we consider a decision maker who experienced the prospect \(\left( 60, 0.25; 30, 0.25, 0\right) \) and is indifferent between saving 28 min for sure and receiving this prospect.
1.2 Appendix 2: Mixed modeling
The likelihood of a series of choices observed by a subject \(i\) writes
Assuming that \(\genfrac(){0.0pt}0{\beta _\mathrm{m}}{\beta _t} \equiv N\left( \genfrac(){0.0pt}0{\overline{\beta _\mathrm{m}}}{\overline{\beta _t}} \begin{bmatrix} \theta _\mathrm{m},&\theta _{\mathrm{m},t}\\ \theta _{\mathrm{m},t}^t,&\theta _t \end{bmatrix} \right) \equiv N\left( \varTheta \right) \) this likelihood becomes:
.
The total likelihood writes
.
The integral of equation (4) does not have a closed form. Therefore, it has has to be simulated.
Considering R draws \(\beta _r\) such that \(\beta _r\sim N\left( \varTheta \right) \), the likelihood \(L_i\left( \varTheta \right) \) can be approximated by its simulated value
Then, the total likelihood can be estimated as
If the number of draws is large enough, the simulated likelihood \(L_{s}\left( \varTheta \right) \) converges to \(L\left( \varTheta \right) \) and the parameters \(\varTheta \) that maximize the \(L\left( \varTheta \right) \) can be obtained by maximizing \(L_{s}\left( \varTheta \right) \).
1.3 Appendix 3: Descriptive statistics on certainty equivalents
1.4 Appendix 4: Distributions of observed frequencies
See Fig. 7.
1.5 Appendix 5: Estimations based on observed frequencies
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Kemel, E., Travers, M. Comparing attitudes toward time and toward money in experience-based decisions. Theory Decis 80, 71–100 (2016). https://doi.org/10.1007/s11238-015-9490-3
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DOI: https://doi.org/10.1007/s11238-015-9490-3