Figure 1.
Trial structure of the experiment.
Participants entered an experimental run of 60 trials, on which they could expect to receive mildly painful electric shock stimuli on each trial, referred to as painful episodes. By default participants could expect to receive a five second stimulus with 14 brief shocks on each trial, however they were provided with a budget of relief at the outset of the experiment, 2400 “milligrammes” (mg) in total. Each 10mg of relief consumed reduced the expected number of shocks in the stimulus train by one, and was hence insufficient to relieve all the shocks in the session. At the start of each trial a screen indicated the number of remaining trials and the remaining supply of relief. Participants then had the opportunity to indicate how much relief they wished to consume on that trial, up to a maximum of 120mg. The relief was then effective on the immediately subsequent painful episode.
Figure 2.
Changes in the instantaneous utility function.
A Linear and concave utility functions. B Simulated optimal consumption paths with no discounting under the two forms of utility function. In each case two sample simulated paths are displayed, to illustrate that, with linear utility there is more than one optimal path. Left panel: under linear utility with no discounting or anticipation all paths which consume the entire budget are equally valued. Consumption is therefore chosen at random from a uniform distribution until the budget is expended. Right panel: concave utility motivates spreading consumption over time.
Figure 3.
Anticipation-discounting functions.
Anticipation-discounting functions are constructed from a linear combination of the conventionally discounted value of an outcome, i.e. its instantaneous anticipation, and the prospective sum of anticipation whilst waiting for the outcome, displayed here for an outcome with positive utility. A Where prospective anticipation (savoring) dominates, the overall value of the outcome decreases as it draws nearer, due to decreasing prospective anticipation. B Where discounting dominates, the overall value of the outcome increases as it draws nearer due to increasing instantaneous anticipation.
Figure 4.
Anticipation-discounting and dynamic utility maximization.
A Four anticipation-discounting functions. From left to right: predominant discounting, no discounting, predominant savoring, discounted savoring. The parameters of each function are displayed on the plot. B Simulated optimal consumption paths under the same four discount functions, with concave utility, U(c) = c0.75 Green circles represent simulated consumption paths for a fully naïve decision-maker (See Main Text). Red circles represent consumption for a fully sophisticated decision-maker. C Plans for future consumption made in the first three time periods for a naïve decision-maker. The red circles indicate planned consumption from the perspective of t = 1, the blue circles from the perspective of t = 2 and the green circles from the perspective of t = 3. Where discounting dominates (left panel), the naïve decision-maker consumes more than planned, where savoring dominates (right hand two panels), the naïve decision-maker consumes less than planned.
Figure 5.
Observed distribution of relief consumption across time.
A Median relief consumption on each trial at the group level is indicated by the solid black circles. Error bars indicate the upper and lower quartiles of consumption. B The distribution of group level consumption. The intensity of the grey bars represents the proportion of the 30 participants included in the analysis choosing to consume each amount of relief on a particular trial. Relief is expressed in units, produced by rounding the raw consumption choices to the nearest 10 units of relief. A tendency to conserve relief is evident from below-average consumption over the first 40 trials and above-average consumption over the last 20 trials. A tendency to spread relief across time is evident from the high proportion of choices to spend 4 units of relief, the mean rounded relief per trial over the whole experimental run. C Consumption paths from six sample participants. Hollow circles denote the consumption choices on each trial. The bold dashed line denotes the mean relief remaining at the start of each trial. These six participants are selected as representative of the key patterns observed. The first two participants (i and ii) on spread relief over time. The subsequent three participants (iii-v) predominantly conserve relief, as evidenced by an increase in the mean relief remaining per trial over time. Participant vi) consumes above the mean relief remaining towards the start of the session and subsequently adjusts consumption downward.
Figure 6.
Relationships between one-off (binary) and dynamic intertemporal choices.
The frequency of choosing sooner pain in the binary intertemporal choice experiment which provides a summary behavioral measure of dread, in both pain (i) and relief (ii) frames (see main text), is plotted against: A the slope of the spending profile in the dynamic consumption task (positive slope indicates saving relief) and B the absolute slope of the spending profile (a measure of deviation from even spreading of relief). Solid lines indicate a linear least-squares fit through the data. There are no significant relationships between the behavioral metrics on the two tasks.
Figure 7.
Optimal consumption paths predicted from anticipation-discounting functions derived from binary choices.
Naïve (green circles) and sophisticated optimal (red circles) paths, derived from binary intertemporal choices in both pain (left column, A) and relief (right column, B) frames with softmax β = 10 and U(c) = c0.75 are overlaid on observed consumption paths (blue circles) for 4 sample participants.
Figure 8.
Fits of the anticipation-discounting model with variable utility and choice randomness.
A The observed distribution of consumption at the group level by participants for whom anticipation-discounting functions derived one-off choice tasks were available (N = 23). Warmer colors indicate that a higher proportion of participants chose to consume that amount of relief on a particular trial. B Group-Level distribution of relief consumption predicted by the optimal model and modifications to it. These plots denote the mean probability across all participants of consuming an amount of relief, ct, on each trial, t, given a vector of the total remaining relief for each participant on each trial, st, st+1, st+2, … sT, at the maximum likelihood parameters, θ, of each model. i) Anticipation-discounting functions derived from one-off pain frame choices, with the softmax temperature, beta, and utility parameters freely fitted. ii) Anticipation-discounting functions derived from one-off relief frame choices, with the softmax temperature, beta, and utility parameters freely fitted. C The proportion of variance explained by each model. Mean predicted consumption levels simulated from the maximum likelihood parameterizations of each model over each 10 trials of the experiment for each participant are plotted against the same metric derived from the observed data.
Figure 9.
A The observed distribution of consumption by all 30 participants included in the analysis. Warmer colors indicate that a higher proportion of participants chose to consume that amount of relief on a particular trial. Black arrows indicate spending zero relief, which becomes more prominent during the middle of the experiment. B Group-Level distribution of relief consumption predicted by alternative heuristic models. These plots denote the mean probability across all participants of consuming an amount of relief, ct, on each trial, t, given a vector of the total remaining relief for each participant on each trial, trial, st, st+1, st+2, … sT, at the maximum likelihood parameterization, θ, of each model. The Direct Action model combines the three key observed behavioral tendencies as heuristics to either spend close zero relief until the mean relief remaining reaches the maximum allowable spend (save-now-spend-later), to spending close to the mean relief remaining per trial (spread-spending) or close to the maximum allowable relief (spend-now-suffer-later). The Income Maximization model extends this model, such that the saving tendency is implemented as the attempt to dynamically maximize the mean remaining relief per trial, over a limited future horizon. This model captures the relatively greater tendency to save relief during the middle of the experiment (as indicated by the black arrows). C The proportion of variance explained by each model. Mean predicted consumption levels simulated from the maximum likelihood parameterizations of each model over each 10 trials of the experiment for each participant are plotted against the same metric derived from the observed data. Least squares fits indicate an R-squared value of 0.56 for the Direct Action model and 0.80 for the Income Maximization model.
Figure 10.
Policy weightings of heuristic models.
A Mean weighting across subjects on each of the three policies (see Methods) on each trial under the maximum likelihood fits of the Direct Action (left) and Income Maximization (right) models. Spend-now-suffer-later has low weighting early in the experiment. Spread-spending has high weightings throughout. For the income maximization model, saving is weighted most highly in the middle part of the experiment. B Box plot showing distribution over participants of policy weightings, averaged over all trials of the experiment. Saving and Spread-Spending heuristics dominate.
Table 1.
Putative mechanisms underlying save-now-spend-later, spend-now-suffer-later and spread-spending heuristics.