Abstract
This paper investigates how individuals evaluate delayed outcomes with risky realization times. Under the discounted expected utility (DEU) model, such evaluations depend only on intertemporal preferences. We obtain several testable hypotheses using the DEU model as a benchmark and test these hypotheses in three experiments. In general, our results show that the DEU model is a poor predictor of intertemporal choice behavior under timing risk. We found that individuals are averse to timing risk and that they evaluate timing lotteries in a rank-dependent fashion. The main driver of timing risk aversion is nothing but probabilistic risk aversion that stems from the nonlinear treatment of probabilities.
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Notes
Throughout this paper, we will refer to those situations where the decision maker has more than one possible delay as risky timing or risky delays since we investigate delays with known (objective) probabilities.
In another study, Leclerc, Schmitt and Dubé (1995) focused on risky decision making for waiting times, examining those situations where time itself is a resource. That is, they were interested in the value of time. In the current study, we are interested in the changes in an outcome’s value as time changes. Thus, our problem is about time preferences rather than the utility for time.
We assume that x is a desirable outcome, that is, the decision maker’s utility u(.) is an increasing function of x.
We will refer to a generalized discounted utility model where the discount function can take any form as a discounted expected utility (DEU) model hereafter.
To see this, consider the following inequalities: \( pD{\left( {t_{1} } \right)} + {\left( {1 - p} \right)}D{\left( {t_{2} } \right)} \geqslant \alpha D{\left( {t_{1} } \right)} + {\left( {1 - \alpha } \right)}D{\left( {t_{2} } \right)} \), since p ≥ α and \( \alpha D{\left( {t_{1} } \right)} + {\left( {1 - \alpha } \right)}D{\left( {t_{2} } \right)} > D{\left( {\alpha t_{1} + {\left( {1 - \alpha } \right)}t_{2} } \right)} \), due to convexity of D(.). These two inequalities together would imply \( pD{\left( {t_{1} } \right)} + {\left( {1 - p} \right)}D{\left( {t_{2} } \right)} - D{\left( {\alpha t_{1} + {\left( {1 - \alpha } \right)}t_{2} } \right)} > 0 \) for all p ≥ α.
At the time of the experiment, 1 New Turkish Lira was approximately 0.625 Euro.
The scenario that we used in the loss domain was similar. In that scenario, participants were asked to imagine that they were going to pay a tax and they could choose between the two timing options.
Our interpretation of the Likert scale is strictly ordinal. We use the scale to allow subjects to express indifference between the two options and to check for within-subject consistency.
Experimental instructions are available from the authors upon request.
We imposed a concave utility function u(x) = x 0.88 to calculate the average monthly discount rates. We used the formula \( {\left( {e^{{ - r_{i} T}} } \right)}X^{{0.88}} = {\text{PV}}^{{0.88}}_{i} \) where T is the number of months, X is the delayed amount, PV i is the present value by participant i and r i is the average monthly discount rate of participant i.
This follows from variability ordering. For L 1, L 2 ≥ 0 such that E[L 1 ] = E[L 2 ], L 1 is more variable than L 2 if and only if E[h(L 1 )] ≥ E[h(L 2 )] for all convex h (Ross 1996).
If L 1 and L 2 are nonnegative random variables with distributions F and G, respectively, then L 1 is more variable than L 2 if and only if \( {\int_a^\infty {{\left( {1 - F{\left( x \right)}} \right)}{\text{d}}x \geqslant } }{\int_a^\infty {{\left( {1 - G{\left( x \right)}} \right)}{\text{d}}x} } \) for all ≥ 0 (see Ross 1996).
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Acknowledgments
The authors would like to thank Onur Boyabatli, Philippe Delquie, participants at the 2006 Workshop on Decision-Making and Utility, the editor and an anonymous referee for their helpful comments and suggestions.
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Onay, S., Öncüler, A. Intertemporal choice under timing risk: An experimental approach. J Risk Uncertainty 34, 99–121 (2007). https://doi.org/10.1007/s11166-007-9005-x
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DOI: https://doi.org/10.1007/s11166-007-9005-x