Fig 1.
Example of predicted effects of choice bundling on preference for larger, later rewards (LLRs) over smaller, sooner rewards (SSRs).
Panel A depicts a choice between: (1) a discrete LLR (LLR1) delivered after a delay (D1; in this example, one year); and (2) a discrete SSR (SSR1), delivered immediately. The corresponding graph to the right illustrates how the discounted value (V) of the LLR decreases hyperbolically according to Eq 1 (in this example, k = 0.003). Individual data points depict the subjective value of the LLR and SSR rewards. Note that the SSR value is not discounted because it is available immediately. Here, Eq 1 predicts preference for the SSR because its subjective value exceeds that of the LLR. Panel B depicts a choice between two bundled reward options, each comprising three rewards: (1) LLR1, LLR2, and LLR3 delivered after D1, D2, and D3, respectively; and (2) SSR1, SSR2, and SSR3 delivered immediately, after D1 and D2, respectively. The graph to the right illustrates how the discounted values (V) of both the LLR and SSR decrease with increasing delay in accord with Eq 1 (k = 0.003, as in Panel A). Individual data points depict the subjective value of individual rewards in their respective bundles, with the inset depicting summed discounted values (Vbunde) of these rewards according to Eq 2. Here, preference for the LLR option is now predicted because its summed value exceeds that of the SSR option.
Fig 2.
Choices presented in the bundle-size 1, 3, and 9 conditions.
The delay to the first LLR in all conditions was D1, with additional rewards in the bundle (where applicable) delivered after D2, D3, etc. These additional delays were equal to D1 multiplied by the order of that delay in the series (e.g., D2 = D1*2, D3 = D1*3, etc.). The first SSR in all conditions was delivered immediately, with additional rewards in the bundle (where applicable) delivered after D1, D2, D3, etc. Note that with this method of arranging delays, the interval between any two rewards in a given bundle was equal to D1. Likewise, any given SSR in a bundle was available D1 units sooner than its paired LLR.
Table 1.
Participant characteristics.
Fig 3.
Mean log indifference delay (ID; ±95% confidence intervals) as a function of bundle size in both ascending and descending order groups combined (left panel) and in individual order groups (right panel). Gray and patterned bands represent 95% confidence intervals around model-predicted effects of bundle size, based on participants’ BS1 indifference delay values (control condition). Linear indifference delay values are scaled on the right y axis to aid in interpretation.
Fig 4.
Mean difference scores (±95% confidence intervals) between observed and predicted log indifference delay (ID).
Depicted are data for the BS 3 and 9 conditions (based on BS1 values) in both order groups combined (left panel), ascending order group (middle panel), and descending order group (right panel). The horizontal gray band reflects the equivalence interval, defined as zero plus or minus 0.1 standard deviations of difference scores.