Comparing attitudes toward time and toward money in experience-based decisions

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.

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.

Table 7 Illustration of the modified bisection method

1.2 Appendix 2: Mixed modeling

The likelihood of a series of choices observed by a subject $i$ writes

$$\begin{aligned} \mathrm{LL}\left( \beta _i\right) =\prod _{j=1}^{K} p_{i,j} \left( c_\mathrm{inf}^{i,j},c_\mathrm{sup}^{i,j}\mid \beta _i\right) \end{aligned}$$

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:

$$\begin{aligned} L_i\left( \varTheta \right) =\int \limits \prod _{j=1}^{K} p \left( c_\mathrm{inf}^{i,j},c_\mathrm{sup}^{i,j} ,\beta \right) \varphi \left( \beta ,\varTheta \right) \mathrm{d}\beta \end{aligned}$$

(4)

.

The total likelihood writes

$$\begin{aligned} L\left( \varTheta \right) =\prod _{i=1}^{N} L_i\left( \varTheta \right) \end{aligned}$$

.

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

$$\begin{aligned} L_{s,i}\left( \varTheta \right) =\frac{1}{R}\sum _{r=1}^R \left( \prod _{j=1}^{K} p_{i,j} \left( c_\mathrm{inf}^{i,j},c_\mathrm{sup}^{i,j}\mid \beta _r\right) \right) . \end{aligned}$$

Then, the total likelihood can be estimated as

$$\begin{aligned} L_{s}\left( \varTheta \right) =\prod _{i=1}^{N} L_{s,i}\left( \varTheta \right) \end{aligned}$$

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

See Tables 8, 9

Table 8 CE under complete information condition

Table 9 CE under incomplete information condition

1.4 Appendix 4: Distributions of observed frequencies

See Fig. 7.

Fig. 7

Mean and standard deviations of observed frequencies

1.5 Appendix 5: Estimations based on observed frequencies

Tables 10, 11.

Table 10 Distribution of RDU parameters for time and money under complete and incomplete information conditions, based on observed frequencies

Table 11 Results of mixed model estimations by the EM algorithm, based on observed frequencies